EXAMPLE 1 Recording and Sorting

Sample values (observations, measurements) should be recorded in the order in which they occur. Sorting, that is, ordering the sample values by size, is done as a first step of investigating properties of the sample and graphing it. Sorting is a standard process on the computer; see Ref. [E35], listed in App. 1.

Super alloys is a collective name for alloys used in jet engines and rocket motors, requiring high temperature (typically 1800°F), high strength, and excellent resistance to oxidation. Thirty specimens of Hastelloy C (nickel-based steel, investment cast) had the tensile strength (in 1000 lb/sq in.), recorded in the order obtained and rounded to integer values,

Sorting gives

EXAMPLE 2 Stem-and-Leaf Plot (Fig. 507)

This is one of the simplest but most useful representations of data. For (1) it is shown in Fig. 507. The numbers in (1) range from 78 to 99; see (2). We divide these numbers into 5 groups, 75–79, 80–84, 85–89, 90–94, 95–99. The integers in the tens position of the groups are 7, 8, 8, 9, 9. These form the stem in Fig. 507. The first leaf is 789, representing 77, 78, 79. The second leaf is 1123344, representing 81, 81, 82, 83, 83, 84, 84. And so on.

The number of times a value occurs is called its absolute frequency. Thus 78 has absolute frequency 1, the value 89 has absolute frequency 5, etc. The column to the extreme left in Fig. 507 shows the cumulative absolute frequencies, that is, the sum of the absolute frequencies of the values up to the line of the leaf. Thus, the number 10 in the second line on the left shows that (1) has 10 values up to and including 84. The number 23 in the next line shows that there are 23 values not exceeding 89, etc. Dividing the cumulative absolute frequencies by n(= 30 in Fig. 507) gives the cumulative relative frequencies0.1, 0.33, 0.76, 0.93, 1.00.

EXAMPLE 3 Histogram (Fig. 508)

For large sets of data, histograms are better in displaying the distribution of data than stem-and-leaf plots. The principle is explained in Fig. 508. (An application to a larger data set is shown in Sec. 25.7). The bases of the rectangles in Fig. 508 are the x-intervals (known as class intervals) 74.5–79.5, 79.5–84.5, 84.5–89.5, 89.5–94.5, 94.5–99.5, whose midpoints (known as class marks) are x = 77, 82, 87, 92, 97, respectively. The height of a rectangle with class mark x is the relative class frequency f_rel(x), defined as the number of data values in that class interval, divided by n (= 30 in our case). Hence the areas of the rectangles are proportional to these relative frequencies, 0.10, 0.23, 0.43, 0.17, 0.07, so that histograms give a good impression of the distribution of data.

EXAMPLE 4 Boxplot. Median. Interquartile Range. Outlier

A boxplot of a set of data illustrates the average size and the spread of the values, in many cases the two most important quantities characterizing the set, as follows.

The average size is measured by the median, or middle quartile, q_M. If the number n of values of the set is odd, then qM is the middlemost of the values when ordered as in (2). If n is even, then qM is the average of the two middlemost values of the ordered set. In (2) we have n = 30 and thus . (In general, will be a fraction if n is even.)

The spread of values can be measured by the range R = x_max − x_min, the largest value minus the smallest one.

Better information on the spread gives the interquartile range IQR = qU − qL. Here qU is the middlemost value (or the average of the two middlemost values) in the data above the median; and qL is the middlemost value (or the average of the two middlemost values) in the data below the median. Hence in (2) we have qU = x₂₃ = 89, qL = x₈ = 83, and IQR = 89 − 83 = 6.

The box in Fig. 509 extends vertically from qL to qU; it has height IQR = 6. The vertical lines below and above the box extend from x_min = 77 to x_max = 99, so that they show R = 22.

Fig. 509. Boxplot of the data set (1)

The line above the box is suspiciously long. This suggests the concept of an outlier, a value that is more than 1.5 times the IQR away from either end of the box; here 1.5 is purely conventional. An outlier indicates that something might have gone wrong in the data collection. In (2) we have 89 + 1.5 IQR = 98, and we regard 99 as an outlier.

EXAMPLE 5 Empirical Rule and Outliers. z-Score

For (1), with and s = 4.8, the three intervals in the Rule are 81.9 x 91.5, 77.1 x 96.3, 72.3 x 101.1 and contain 73% (22 values remain, 5 are too small, and 5 too large), 93% (28 values, 1 too small, and 1 too large), and 100%, respectively.

If we reduce the sample by omitting the outlier 99, mean and standard deviation reduce to , approximately, and the percentage values become 67% (5 and 5 values outside), 93% (1 and 1 outside), and 100%

Finally, the relative position of a value x in a set of mean and standard deviation s can be measured by the z-score

This is the distance of x from the mean measured in multiples of s. For instance, z(83) = (83 − 86.7)/4.8 = −0.77. This is negative because 83 lies below the mean. By the Empirical Rule, the extreme z-values are about −3 and 3.

PROBLEM SET 24.1

1–10 DATA REPRESENTATIONS

Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:

1. Length of nails [mm]

   

2. Phone calls per minute in an office between 9.00 A.M. and A.M.

   

3. Systolic blood pressure of 15 female patients of ages 20–22

   

4. Iron content [%] of 15 specimens of hermatite (Fe₂O₃)

   

5. Weight of filled bags [g] in an automatic filling

   

6. Gasoline consumption [miles per gallon, rounded] of six cars of the same model under similar conditions

   

7. Release time [sec] of a relay

   

8. Foundrax test of Brinell hardness (2.5 mm steel ball, 62.5 kg load, 30 sec) of 20 copper plates (values in kg/mm²)

   

9. Efficiency [%] of seven Voith Francis turbines of runner diameter 2.3 m under a head range of 185 m

   

10. −0.51 0.12 −0.47 0.95 0.25 −0.18 −0.54

11–16 AVERAGE AND SPREAD

Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.

• 11. For the data in Prob. 1
• 12. For the phone call data in Prob. 2
• 13. For the medical data in Prob. 3
• 14. For the iron contents in Prob. 4
• 15. For the release times in Prob. 7
• 16. For the Brinell hardness data in Prob. 8
• 17. Outlier, reduced data. Calculate s for the data 4 1 3 10 2. Then reduce the data by deleting the outlier and calculate s. Comment.
• 18. Outlier, reduction. Do the same tasks as in Prob. 17 for the hardness data in Prob. 8.
• 19. Construct the simplest possible data with but q_M = 0. What is the point of this problem?
• 20. Mean. Prove that must always lie between the smallest and the largest data values.

EXAMPLES 1–6 Random Experiments. Sample Spaces

1. Inspecting a lightbulb. S = {Defective, Nondefective}.
2. Rolling a die. S = {1, 2, 3, 4, 5, 6}.
3. Measuring tensile strength of wire. S the numbers in some interval.
4. Measuring copper content of brass. s: 50% to 90%, say.
5. Counting daily traffic accidents in New York. S the integers in some interval.
6. Asking for opinion about a new car model. S = {Like, Dislike, Undecided}.

EXAMPLE 7 Events

In (2), events are A = {1, 3, 5} (“Odd number”), B = {2, 4, 6} (“Even number”), C = {5, 6}. etc. Simple events are {1}, {2}, …, {6}.

EXAMPLE 8 Unions and Intersections of 3 Events

In rolling a die, consider the events

Then A ∩ B = {4, 5}, B ∩ C = {2, 4}, C ∩ A = {4, 6}, A ∩ B ∩ C = {4}. Can you sketch a Venn diagram of this? Furthermore, A ∪ B = s, hence A ∪ B = s, hence A ∪ B ∪ C = s (why?).

PROBLEM SET 24.2

1–12 SAMPLE SPACES, EVENTS

Graph a sample space for the experiments:

1. Drawing 3 screws from a lot of right-handed and left-handed screws
2. Tossing 2 coins
3. Rolling 2 dice
4. Rolling a die until the first Six appears
5. Tossing a coin until the first Head appears
6. Recording the lifetime of each of 3 lightbulbs
7. Recording the daily maximum temperature X and the daily maximum air pressure Y at Times Square in New York
8. Choosing a committee of 2 from a group of 5 people
9. Drawing gaskets from a lot of 10, containing one defective D, unitil D is drawn, one at a time and assuming sampling without replacement, that is, gaskets drawn are not returned to the lot. (More about this in Sec. 24.6)
10. In rolling 3 dice, are the events A: Sum divisible by 3 and B: Sum divisible by 5 mutually exclusive?
11. Answer the questions in Prob. 10 for rolling 2 dice.
12. List all 8 subsets of the sample space S = {a, b, c}.
13. In Prob. 3 circle and mark the events A: Faces are equal, B: Sum of faces less than 5, A ∪ B, A ∩ B, A^c, B^c.
14. In drawing 2 screws from a lot of right-handed and left-handed screws, let A, B, C, D mean at a least 1 right-handed, at least 1 left-handed, 2 right-handed, 2 left-handed, respectively. Are A and B mutually exclusive? C and D?

15–20 VENN DIAGRAMS

• 15. In connection with a trip to Europe by some students, consider the events P that they see Paris, G that they have a good time, and M that they run out of money, and describe in words the events 1, …, 7 in the diagram.

  

  Problem 15

• 16. Show that, by the definition of complement, for any subset A of a sample space S.

  

• 17. Using a Venn diagram, show that A ⊆ B if and only if A ∪ B = B.
• 18. Using a Venn diagram, show that A ⊆ B if and only if A ∩ B = A.
• 19. (De Morgan's laws) Using Venn diagrams, graph and check De Morgan's laws

  

• 20. Using Venn diagrams, graph and check the rules

  

DEFINITION 1 First Definition of Probability

If the sample space S of an experiment consists of finitely many outcomes (points) that are equally likely, then the probability P(A) of an event A is

EXAMPLE 1 Fair Die

In rolling a fair die once, what is the probability P(A) of A of obtaining a 5 or a 6? The probability of B:“Even number”?

Solution. The six outcomes are equally likely, so that each has probability 1/6. Thus P(A) = 2/6 = 1/3 because A = {5, 6} has 2 points, and P(B) = 3/6 = 1/2.

DEFINITION 2 General Definition of Probability

Given a sample space S, with each event A of S (subset of S) there is associated a number P(A), called the probability of A, such that the following axioms of probability are satisfied.

1. For every A in S,

2. The entire sample space S has the probability

3. For mutually exclusive events A and B (A ∩ B) = Ø; see Sec. 24.2),

If S is infinite (has infinitely many points), Axiom 3 has to be replaced by 3′. For mutually exclusive events A₁, A₂, …,

THEOREM 1 Complementation Rule

For an event A and its complement A^c in a sample space S,

PROOF

By the definition of complement (Sec. 24.2), we have S = A ∪ A^c and A ∩ A^c = Ø. Hence by Axioms 2 and 3,

EXAMPLE 2 Coin Tossing

Five coins are tossed simultaneously. Find the probability of the event A: At least one head turns up. Assume that the coins are fair.

Solution. Since each coin can turn up heads or tails, the sample space consists of 2⁵ = 32 outcomes. Since the coins are fair, we may assign the same probability (1/32) to each outcome. Then the event A^c (No heads turn up) consists of only 1 outcome. Hence P(A^c) = 1/32, and the answer is P(A) = 1 − P(A^c) = 31/32.

THEOREM 2 Addition Rule for Mutually Exclusive Events

For mutually exclusive events A₁, …, A_m in a sample space S,

EXAMPLE 3 Mutually Exclusive Events

If the probability that on any workday a garage will get 10–20, 21–30, 31–40, over 40 cars to service is 0.20, 0.35, 0.25, 0.12, respectively, what is the probability that on a given workday the garage gets at least 21 cars to service?

Solution. Since these are mutually exclusive events, Theorem 2 gives the answer 0.35 + 0.25 + 0.12 = 0.72. Check this by the complementation rule.

THEOREM 3 Addition Rule for Arbitrary Events

For events A and B in a sample space,

PROOF

C, D, E in Fig. 512 make up A ∪ B and are mutually exclusive (disjoint). Hence by Theorem 2,

This gives (9) because on the right P(C) + P(D) = P(A) by Axiom 3 and disjointness; and P(E) = P(B) − P(D) = P(B) − P(A ∩ B), also by Axiom 3 and disjointness.

EXAMPLE 4 Union of Arbitrary Events

In tossing a fair die, what is the probability of getting an odd number or a number less than 4?

Solution. Let A be the event “Odd number” and B the event “Number less than 4.” Then Theorem 3 gives the answer

because A ∩ B = “Odd number less than 4” = {1, 3}.

THEOREM 4 Multiplication Rule

If A and B are events in a sample space S and P(A) ≠ 0, P(B) ≠ 0, then

EXAMPLE 5 Multiplication Rule

In producing screws, let A mean “screw too slim” and B “screw too short.” Let P(A) = 0.1 and let the conditional probability that a slim screw is also too short be P(B|A) = 0.2. What is the probability that a screw that we pick randomly from the lot produced will be both too slim and too short?

Solution. P(A ∩ B) = P(A)P(B|A) = 0.1 · 0.2 = 0.02 = 2%, by Theorem 4.

EXAMPLE 6 Sampling With and Without Replacement

A box contains 10 screws, three of which are defective. Two screws are drawn at random. Find the probability that neither of the two screws is defective.

Solution. We consider the events

Clearly, because 7 of the 10 screws are nondefective and we sample at random, so that each screw has the same probability of being picked. If we sample with replacement, the situation before the second drawing is the same as at the beginning, and . The events are independent, and the answer is

If we sample without replacement, then , as before. If A has occurred, then there are 9 screws left in the box, 3 of which are defective. Thus , and Theorem 4 yields the answer

Is it intuitively clear that this value must be smaller than the preceding one?

PROBLEM SET 24.3

1. In rolling 3 fair dice, what is the probability of obtaining a sum not greater than 16?
2. In rolling 2 fair dice, what is the probability of a sum greater than 3 but not exceeding 6?
3. Three screws are drawn at random from a lot of 100 screws, 10 of which are defective. Find the probability of the event that all 3 screws drawn are nondefective, assuming that we draw (a) with replacement, (b) without replacement.
4. In Prob. 3 find the probability of E: At least 1 defective (i) directly, (ii) by using complements; in both cases (a) and (b).
5. If a box contains 10 left-handed and 20 right-handed screws, what is the probability of obtaining at least one right-handed screw in drawing 2 screws with replacement?
6. Will the probability in Prob. 5 increase or decrease if we draw without replacement. First guess, then calculate.
7. Under what conditions will it make practically no difference whether we sample with or without replacement?
8. If a certain kind of tire has a life exceeding 40,000 miles with probability 0.90, what is the probability that a set of these tires on a car will last longer than 40,000 miles?
9. If we inspect photocopy paper by randomly drawing 5 sheets without replacement from every pack of 500, what is the probability of getting 5 clean sheets although 0.4% of the sheets contain spots?
10. Suppose that we draw cards repeatedly and with replacement from a file of 100 cards, 50 of which refer to male and 50 to female persons. What is the probability of obtaining the second “female” card before the third “male” card?
11. A batch of 200 iron rods consists of 50 oversized rods, 50 undersized rods, and 100 rods of the desired length. If two rods are drawn at random without replacement, what is the probability of obtaining (a) two rods of the desired length, (b) exactly one of the desired length, (c) none of the desired length?
12. If a circuit contains four automatic switches and we want that, with a probability of 99%, during a given time interval the switches to be all working, what probability of failure per time interval can we admit for a single switch?
13. A pressure control apparatus contains 3 electronic tubes. The apparatus will not work unless all tubes are operative. If the probability of failure of each tube during some interval of time is 0.04, what is the corresponding probability of failure of the apparatus?
14. Suppose that in a production of spark plugs the fraction of defective plugs has been constant at 2% over a long time and that this process is controlled every half hour by drawing and inspecting two just produced. Find the probabilities of getting (a) no defectives, (b) 1 defective, (c) 2 defectives. What is the sum of these probabilities?
15. What gives the greater probability of hitting at least once: (a) hitting with probability 1/2 and firing 1 shot, (b) hitting with probability 1/4 and firing 2 shots, (c) hitting with probability 1/8 and firing 4 shots? First guess.
16. You may wonder whether in (16) the last relation follows from the others, but the answer is no. To see this, imagine that a chip is drawn from a box containing 4 chips numbered 000, 011, 101, 110, and let A, B, C be the events that the first, second, and third digit, respectively, on the drawn chip is 1. Show that then the first three formulas in (16) hold but the last one does not hold.
17. Show that if B is a subset of A, then P(B) P(A).
18. Extending Theorem 4, show that P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B).
19. Make up an example similar to Prob. 16, for instance, in terms of divisibility of numbers.

THEOREM 1 Permutations

(a)Different things. The number of permutations of n different things taken all at a time is

(b)Classes of equal things. If n given things can be divided into c classes of alike things differing from class to class, then the number of permutations of these things taken all at a time is

Where n_j is the number of things in the jth class.

PROOF

(a) There are n choices for filling the first place in the row. Then n − 1 things are still available for filling the second place, etc.

(b) n₁ alike things in class 1 make n₁! permutations collapse into a single permutation (those in which class 1 things occupy the same n₁ positions), etc., so that (2) follows from (1).

EXAMPLE 1 Illustration of Theorem 1(b)

If a box contains 6 red and 4 blue balls, the probability of drawing first the red and then the blue balls is

THEOREM 2 Permutations

The number of different permutations of n different things taken k at a time without repetitions is

and with repetitions is

EXAMPLE 2 Illustration of Theorem 2

In an encrypted message the letters are arranged in groups of five letters, called words. From (3b) we see that the number of different such words is

From (3a) it follows that the number of different such words containing each letter no more than once is

THEOREM 3 Combinations

The number of different combinations of n different things taken, k at a time, without repetitions, is

and the number of those combinations with repetitions is

PROOF

The statement involving (4a) follows from the first part of Theorem 2 by noting that there are k! permutations of k things from the given n things that differ by the order of the elements (see Theorem 1), but there is only a single combination of those k things of the type characterized in the first statement of Theorem 3. The last statement of Theorem 3 can be proved by induction (see Team Project 14).

EXAMPLE 3 Illustration of Theorem 3

The number of samples of five lightbulbs that can be selected from a lot of 500 bulbs is [see (4a)]

EXAMPLE 4 Stirling Formula

PROBLEM SET 24.4

Note the large numbers in the answers to some of these problems, which would make counting cases hopeless!

1. In how many ways can a company assign 10 drivers to n buses, one driver to each bus and conversely?
2. List (a) all permutations, (b) all combinations without repetitions, (c) all combinations with repetitions, of 5 letters a, e, i, o, u taken 2 at a time.
3. If a box contains 4 rubber gaskets and 2 plastic gaskets, what is the probability of drawing (a) first the plastic and then the rubber gaskets, (b) first the rubber and then the plastic ones? Do this by using a theorem and checking it by multiplying probabilities.
4. An urn contains 2 green, 3 yellow, and 5 red balls. We draw 1 ball at random and put it aside. Then we draw the next ball, and so on. Find the probability of drawing at first the 2 green balls, then the 3 yellow ones, and finally the red ones.
5. In how many different ways can we select a committee consisting of 3 engineers, 2 physicists, and 2 computer scientists from 10 engineers, 5 physicists, and 6 computer scientists? First guess.
6. How many different samples of 4 objects can we draw from a lot of 50?
7. Of a lot of 10 items, 2 are defective. (a) Find the number of different samples of 4. Find the number of samples of 4 containing (b) no defectives, (c) 1 defective, (d) 2 defectives.
8. Determine the number of different bridge hands. (A bridge hand consists of 13 cards selected from a full deck of 52 cards.)
9. In how many different ways can 6 people be seated at a round table?
10. If a cage contains 100 mice, 3 of which are male, what is the probability that the 3 male mice will be included if 10 mice are randomly selected?
11. How many automobile registrations may the police have to check in a hit-and-run accident if a witness reports KDP7 and cannot remember the last two digits on the license plate but is certain that all three digits were different?
12. If 3 suspects who committed a burglary and 6 innocent persons are lined up, what is the probability that a witness who is not sure and has to pick three persons will pick the three suspects by chance? That the witness picks 3 innocent persons by chance?
13. CAS PROJECT. Stirling formula. (a) Using (7), compute approximate values of n! for n = 1, …, 20.
    • (b) Determine the relative error in (a). Find an empirical formula for that relative error.
    • (c) An upper bound for that relative error is e^1/12n − 1. Try to relate your empirical formula to this.
    • (d) Search through the literature for further information on Stirling's formula. Write a short eassy about your findings, arranged in logical order and illustrated with numeric examples.
14. 14. TEAM PROJECT. Permutations, Combinations.
    1. Prove Theorem 2.
    2. Prove the last statement of Theorem 3.
    3. Derive (11) from (8).
    4. By the binomial theorem,

       

       so that a^kb^n−k has the coefficient . Can you conclude this from Theorem 3 or is this a mere coincidence?

    5. Prove (14) by using the binomial theorem.
    6. Collect further formulas for binomial coefficients from the literature and illustrate them numerically.
15. 15. Birthday problem. What is the probability that in a group of 20 people (that includes no twins) at least two have the same birthday, if we assume that the probability of having birthday on a given day is 1/365 for every day. First guess. Hint. Consider the complementary event.

DEFINITION Random Variable

A random variable X is a function defined on the sample space S of an experiment. Its values are real numbers. For every number a the probability

with which X assumes a is defined. Similarly, for any interval I the probability

with which X assumes any value in I is defined.

EXAMPLE 1 Probability Function and Distribution Function

Figure 513 shows the probability function f(x) and the distribution function F(x) of the discrete random variable

X has the possible values x = 1, 2, 3, 4, 5, 6 with probability 1/6 each. At these x the distribution function has upward jumps of magnitude 1/6. Hence from the graph of f(x) we can construct the graph of F(x) and conversely.

In Figure 513 (and the next one) at each jump the fat dot indicates the function value at the jump!

EXAMPLE 2 Probability Function and Distribution Function

The random variable X = Sum of the two numbers two fair dice turn up is discrete and has the possible values 2 (= 1 + 1), 3, 4, …, 12 (= 6 · 6). There are 6 · 6 = 36 equally likely outcomes (1, 1) (1, 2), …, (6, 6), where the first number is that shown on the first die and the second number that on the other die. Each such outcome has probability 1/36. Now X = 2 occurs in the case of the outcome (1, 1); X = 3 in the case of the two outcomes (1, 2) and (2, 1); X = 4 in the case of the three outcomes (1, 3), (2, 2), (3, 1); and so on. Hence f(x) = P(X = x) and F(x) = P(X x) have the values

Figure 514 shows a bar chart of this function and the graph of the distribution function, which is again a step function, with jumps (of different height!) at the possible values of X.

EXAMPLE 3 Illustration of Formula (5)

In Example 2, compute the probability of a sum of at least 4 and at most 8.

Solution. .

EXAMPLE 4 Waiting Time Problem. Countably Infinite Sample Space

In tossing a fair coin, let X = Number of trials until the first head appears. Then, by independence of events (Sec. 24.3),

and in general , n = 1, 2, …. Also, (6) can be confirmed by the sum formula for the geometric series,

EXAMPLE 5 Continuous Distribution

Let X have the density function f(x) = 0.75(1 − x)² if −1 x 1 and zero otherwise. Find the distribution function. Find the probabilities and . Find x such that P(X x) = 0.95.

Solution. From (7) we obtain F(x) = 0 if x −1,

and F(x) = 1 if x > 1. From this and (9) we get

(because for a continuous distribution) and

(Note that the upper limit of integration is 1, not 2. Why?) Finally,

Algebraic simplification gives 3x − x³ = 1.8. A solution is x = 0.73, approximately.

Sketch f(x) and mark , and 0.73, so that you can see the results (the probabilities) as areas under the curve. Sketch also F(x).

PROBLEM SET 24.5

1. Graph the probability function f(x) = kx² (x = 1, 2, 3, 4, 5; k suitable) and the distribution function.
2. Graph the density function f(x) = kx² (0 x 5; k suitable) and the distribution function.
3. Uniform distribution. Graph f and F when the density of X is f(x) = k = const if −2 x 2 and 0 else-where. Find P(0 X 2).
4. In Prob. 3 find c and such that P(−c < X < c) = 95% and P(0 < X < ) = 95%.
5. Graph f and F when , . Can f have further positive values?
6. A box contains 4 right-handed and 6 left-handed screws. Two screws are drawn at random without replacement. Let X be the number of left-handed screws drawn. Find the probabilities P(X = 0), P(X = 1), P(X = 2), P(1 < X 2), P(X 1), P(X 1), P(X > 1), and P(0.5 < X < 10).
7. Let X be the number of years before a certain kind of pump needs replacement. Let X have the probability function f(x) = kx³, x = 0, 1, 2, 3, 4, Find k. Sketch f and F.
8. Graph the distribution function F(x) = 1 − e^−3x if x > 0, F(x) = 0 if x 0, and the density f(x). Find x such that F(x) = 0.9.
9. Let X [millimeters] be the thickness of washers. Assume that X has the density f(x) = kx if 0.9 < x < 1.1 and 0 otherwise. Find k. What is the probability that a washer will have thickness between 0.95 mm and 1.05 mm?
10. If the diameter X of axles has the density f(x) = k if 119.9 x 120.1 and 0 otherwise, how many defectives will a lot of 500 axles approximately contain if defectives are axles slimmer than 119.91 or thicker than 120.09?
11. Find the probability that none of three bulbs in a traffic signal will have to be replaced during the first 1500 hours of operation if the lifetime X of a bulb is a random variable with the density f(x) = 6[0.25 − (x − 1.5)²] when 1 x 2 and f(x) = 0 otherwise, where x is measured in multiples of 1000 hours.
12. Let X be the ratio of sales to profits of some company. Assume that X has the distribution function F(x) = 0 if x < 2, F(x) = (x² − 4)/5 if 2 x < 3, F(x) = 1 if x 3. Find and sketch the density. What is the probability that X is between 2.5 (40% profit) and 5 (20% profit)?
13. Suppose that in an automatic process of filling oil cans, the content of a can (in gallons) is Y = 100 + X, where X is a random variable with density f(x) = 1 − |x| when |x| 1 and 0 when |x| > 1. Sketch f(x) and F(x). In a lot of 1000 cans, about how many will contain 100 gallons or more? What is the probability that a can will contain less than 99.5 gallons? Less than 99 gallons?
14. Find the probability function of X = Number of times a fair die is rolled until the first Six appears and show that it satisfies (6).
15. Let X be a random variable that can assume every real value. What are the complements of the events X b, X < b, X c, X > c, b X c, b < X c?

EXAMPLE 1 Mean and Variance

The random variable X = Number of heads in a single toss of a fair coin has the possible values X = 0 and X = 1 with probabilities and . From (la) we thus obtain the mean and (2a) yields the variance

EXAMPLE 2 Uniform Distribution. Variance Measures Spread

The distribution with the density

and f = 0 otherwise is called the uniform distribution on the interval a < x < b. From (1b) (or from Theorem 1, below) we find that μ = (a + b)/2, and (2b) yields the variance

Figure 516 illustrates that the spread is large if and only if σ² is large.

THEOREM 1 Mean of a Symmetric Distribution

If a distribution is symmetric with respect to x = c, that is, f(c − x) = f(c + x), then μ = c. (Examples 1 and 2 illustrate this.)

THEOREM 2 Transformation of Mean and Variance

(a) If a random variable X has mean μ and variance σ², then the random variable

has the mean μ* and variance σ*², where

(b) In particular, the standardized random variable Z corresponding to X, given by

has the mean 0 and the variance 1.

PROOF

We prove (5) for a continuous distribution. To a small interval I of length Δx on the x-axis there corresponds the probability f(x)Δx [approximately; the area of a rectangle of base Δx and height f(x)]. Then the probability f(x)Δx must equal that for the corresponding interval on the x*-axis, that is, f*(x*)Δx*, where f* is the density of X* and Δx* is the length of the interval on the x*-axis corresponding to I. Hence for differentials we have f*(x*) dx* = f(x)dx. Also, x* = a₁ + a₂x by (4), so that (1b) applied to X* gives

On the right the first integral equals 1, by (10) in Sec. 24.5. The second intergral is μ This proves (5) for μ*. It implies

From this and (2) applied to X*, again using f*(x*) dx* = f(x)dx, we obtain the second formula in (5),

For a discrete distribution the proof of (5) is similar.

Choosing a₁ = −μ/σ and a₂ = 1/σ we obtain (6) from (4), writing X* = Z. For these a₁, a₂ formula (5) gives μ* = 0 and σ^*2 = 1, as claimed in (b).

PROBLEM SET 24.6

1–8 MEAN, VARIANCE

Find the mean and variance of the random variable X with probability function or density f(x).

1. f(x) = kx (0 x 2, k suitable)
2. X = Number a fair die turns up
3. Uniform distribution on [0, 2π]
4. 
5. f(x) = 4e^−4x (x 0)
6. f(x) = k(1 − x²) if and 0 otherwise
7. f(x) = Ce^−x/2 (x = 0)
8. X = Number of times a fair coin is flipped until the first Head appears. (Calculate μ only.)
9. If the diameter X [cm] of certain bolts has the density f(x) = k(x − 0.9)(1.1 − x) for 0.9 < x < 1.1 and 0 for other x, what are k, μ and σ²? Sketch f(x).
10. If, in Prob. 9, a defective bolt is one that deviates from 1.00 cm by more than 0.06 cm, what percentage of defectives should we expect?
11. For what choice of the maximum possible deviation from 1.00 cm shall we obtain defectives in Probs. 9 and 10?
12. What total sum can you expect in rolling a fair die 20 times? Do the experiment. Repeat it a number of times and record how the sum varies.
13. What is the expected daily profit if a store sells X air conditioners per day with probability f(10) = 0.1, f(11) = 0.3, f(12) = 0.4, f(13) = 0.2 and the profit per conditioner is $55?
14. Find the expectation of g(X) = X², where X is uniformly distributed on the interval −1 x 1.
15. A small filling station is supplied with gasoline every Saturday afternoon. Assume that its volume X of sales in ten thousands of gallons has the probability density f(x) = 6x(1 − x) if 0 x 1 and 0 otherwise. Determine the mean, the variance, and the standardized variable.
16. What capacity must the tank in Prob. 15 have in order that the probability that the tank will be emptied in a given week be 5%?
17. James rolls 2 fair dice, and Harry pays k cents to James, where k is the product of the two faces that show on the dice. How much should James pay to Harry for each game to make the game fair?
18. What is the mean life of a lightbulb whose life X [hours] has the density f(x) = 0.001e^−0.001x (x 0)?
19. Let X be discrete with probability function Find the expectation of X³.
20. TEAM PROJECT. Means, Variances, Expectations.
    1. Show that E(X − μ) = 0, σ² = E(X²) − μ².
    2. Prove (10)–(12).
    3. Find all the moments of the uniform distribution on an interval a x b.
    4. The skewness γ of a random variable X is defined by

       

       Show that for a symmetric distribution (whose third central moment exists) the skewness is zero.

    5. Find the skewness of the distribution with density f(x) = xe^−x when x > 0 and f(x) = 0 otherwise. Sketch f(x).
    6. Calculate the skewness of a few simple discrete distributions of your own choice.
    7. Find a nonsymmetric discrete distribution with 3 possible values, mean 0, and skewness 0.

EXAMPLE 1 Binomial Distribution

Compute the probability of obtaining at least two “Six” in rolling a fair die 4 times.

Solution. The event “At least two ‘Six’” occurs if we obtain 2 or 3 or 4 “Six.” Hence the answer is

EXAMPLE 2 Poisson Distribution

If the probability of producing a defective screw is p = 0.01, what is the probability that a lot of 100 screws will contain more than 2 defectives?

Solution. The complementary event is A^c: Not more than 2 defectives. For its probability we get, from the binomial distribution with mean μ = np = 1, the value [see (2)]

Since p is very small, we can approximate this by the much more convenient Poisson distribution with mean μ = np = 100 · 0.01 = 1, obtaining [see (5)]

Thus p(A) = 8.03%. Show that the binomial distribution gives p(A) = 7.94%, so that the Poisson approximation is quite good.

EXAMPLE 3 Parking Problems. Poisson Distribution

If on the average, 2 cars enter a certain parking lot per minute, what is the probability that during any given minute 4 or more cars will enter the lot?

Solution. To understand that the Poisson distribution is a model of the situation, we imagine the minute to be divided into very many short time intervals, let p be the (constant) probability that a car will enter the lot during any such short interval, and assume independence of the events that happen during those intervals. Then we are dealing with a binomial distribution with very large n and very small p, which we can approximate by the Poisson distribution with

because 2 cars enter on the average. The complementary event of the event “4 cars or more during a given minute” is “3 cars or fewer enter the lot” and has the probability

Answer: 14.3%. (Why did we consider that complement?)

EXAMPLE 4 Sampling with and without Replacement

We want to draw random samples of two gaskets from a box containing 10 gaskets, three of which are defective. Find the probability function of the random variable X = Number of defectives in the sample.

Solution. We have N = 10, M = 3, N − M = 7, n = 2. For sampling with replacement, (7) yields

For sampling without replacement we have to use (8), finding

PROBLEM SET 24.7

1. Mark the positions of μ in Fig. 517. Comment.
2. Graph (2) for n = 8 as in Fig. 517 and compare with Fig. 517.
3. In Example 3, if 5 cars enter the lot on the average, what is the probability that during any given minute 6 or more cars will enter? First guess. Compare with Example 3.
4. How do the probabilities in Example 4 of the text change if you double the numbers: drawing 4 gaskets from 20, 6 of which are defective? First guess.
5. Five fair coins are tossed simultaneously. Find the probability function of the random variable X = Number of heads and compute the probabilities of obtaining no heads, precisely 1 head, at least 1 head, not more than 4 heads.
6. Suppose that 4% of steel rods made by a machine are defective, the defectives occurring at random during production. If the rods are packaged 100 per box, what is the Poisson approximation of the probability that a given box will contain x = 0, 1, …, 5 defectives?
7. Let X be the number of cars per minute passing a certain point of some road between 8 A.M. and 10 A.M. on a Sunday. Assume that X has a Poisson distribution with mean 5. Find the probability of observing 4 or fewer cars during any given minute.
8. Suppose that a telephone switchboard of some company on the average handles 300 calls per hour, and that the board can make at most 10 connections per minute. Using the Poisson distribution, estimate the probability that the board will be overtaxed during a given minute. (Use Table A6 in App. 5 or your CAS.)
9. Rutherford–Geiger experiments. In 1910, E. Rutherford and H. Geiger showed experimentally that the number of alpha particles emitted per second in a radioactive process is a random variable X having a Poisson distribution. If X has mean 0.5, what is the probability of observing two or more particles during any given second?
10. Let p = 2% be the probability that a certain type of lightbulb will fail in a 24-hour test. Find the probability that a sign consisting of 15 such bulbs will burn 24 hours with no bulb failures.
11. Guess how much less the probability in Prob. 10 would be if the sign consisted of 100 bulbs. Then calculate.
12. Suppose that a certain type of magnetic tape contains, on the average, 2 defects per 100 meters. What is the probability that a roll of tape 300 meters long will contain (a) x defects, (b) no defects?
13. Suppose that a test for extrasensory perception consists of naming (in any order) 3 cards randomly drawn from a deck of 13 cards. Find the probability that by chance alone, the person will correctly name (a) no cards, (b) 1 card, (c) 2 cards, (d) 3 cards.
14. If a ticket office can serve at most 4 customers per minute and the average number of customers is 120 per hour, what is the probability that during a given minute customers will have to wait? (Use the Poisson distribution, Table 6 in Appendix 5.)
15. Suppose that in the production of 60-ohm radio resistors, nondefective items are those that have a resistance between 58 and 62 ohms and the probability of a resistor's being defective is 0.1%. The resistors are sold in lots of 200, with the guarantee that all resistors are nondefective. What is the probability that a given lot will violate this guarantee? (Use the Poisson distribution.)
16. TEAM PROJECT. Moment Generating Function. The moment generating function G(t) is defined by

    

    or

    

    where X is a discrete or continuous random variable, respectively.

    1. Assuming that termwise differentiation and differentiation under the integral sign are permissible, show that E(X^k) = G^(k)(0), where G^(k) = d^kG/dt^k, in particular, μ = G′(0).
    2. Show that the binomial distribution has the moment generating function

       

    3. Using (b), prove (3).
    4. Prove (4).
    5. Show that the Poisson distribution has the moment generating function and prove (6).
    6. Prove

       Using this, prove (9).

17. 17. Multinomial distribution. Suppose a trial can result in precisely one of k mutually exclusive events A₁, …, A_k with probabilities p₁, … p_k, respectively, p₁ + … + p_k = 1. where Suppose that n independent trials are performed. Show that the probability of getting x₁ A₁”s, …, x_k A_k’s is

    

    where 0 x_j n, j = 1, …, k, and x₁ + … + x_k = n. The distribution having this probability function is called the multinomial distribution.

18. A process of manufacturing screws is checked every hour by inspecting n screws selected at random from that hour's production. If one or more screws are defective, the process is halted and carefully examined. How large should n be if the manufacturer wants the probability to be about 95% that the process will be halted when 10% of the screws being produced are defective? (Assume independence of the quality of any screw from that of the other screws.)

THEOREM 1 Use of the Normal Table A7 in App. 5

The distribution function F(x) of the normal distribution with any μ and σ [see (2)] is related to the standardized distribution function Φ(z) in (3) by the formula

PROOF

Comparing (2) and (3) we see that we should set

as the new upper limit of integration. Also ν − μ = σu, thus dν = σ du. Together, since σ drops out,

THEOREM 2 Normal Probabilities for Intervals

The probability that a normal random variable X with mean μ and standard deviation σ assume any value in an interval a < x b is

PROOF

Formula (2) in Sec. 24.5 gives the first equality in (5), and (4) in this section gives the second equality.

EXAMPLE 1 Reading Entries from Table A7

If X is standardized normal (so that μ = 0, σ = 1), then

EXAMPLE 2 Probabilities for Given Intervals, Table A7

Let X be normal with mean 0.8 and variance 4 (so that σ = 2). Then by (4) and (5)

or, if you like it better, (similarly in the other cases)

EXAMPLE 3 Unknown Values c for Given Probabilities, Table A8

Let X be normal with mean 5 and variance 0.04 (hence standard deviation 0.2). Find c or k corresponding to the given probability

EXAMPLE 4 Defectives

In a production of iron rods let the diameter X be normally distributed with mean 2 in. and standard deviation 0.008 in.

1. What percentage of defectives can we expect if we set the tolerance limits at 2 ± 0.02 in.?
2. How should we set the tolerance limits to allow for 4% defectives?

Solution. (a) because from (5) and Table A7 we obtain for the complementary event the probability

(b) 2 ± 0.0164 because, for the complementary event, we have

or

so that Table A8 gives

THEOREM 3 Limit Theorem of De Moivre and Laplace

For large n,

Here f is given by (8). The function

is the density of the normal distribution with mean μ = np and variance σ² = npq (the mean and variance of the binomial distribution). The symbol ~ (read asymptotically equal) means that the ratio of both sides approaches 1 as n approaches ∞. Furthermore, for any nonnegative integers a and b (> a),

PROBLEM SET 24.8

1. Let X be normal with mean 10 and variance 4. Find P(X > 12), P(X < 10), P(X < 11), P(9 < X < 13).
2. Let X be normal with mean 105 and variance 25. Find P(X 112.5), P(x > 100), P(110.5) < X < 111.25).
3. Let X be normal with mean 50 and variance 9. Determine c such that P(X < c) = 5%, P(X > c) = 1%, P(50 − c < X < 50 + c) = 50%.
4. Let X be normal with mean 3.6 and variance 0.01. Find c such that P(X c) = 50%, P(X > c) = 10%, P(−c < X − 3.6 c) = 99.9%.
5. If the lifetime X of a certain kind of automobile battery is normally distributed with a mean of 5 years and a standard deviation of 1 year, and the manufacturer wishes to guarantee the battery for 4 years, what percentage of the batteries will he have to replace under the guarantee?
6. If the standard deviation in Prob. 5 were smaller, would that percentage be larger or smaller?
7. A manufacturer knows from experience that the resistance of resistors he produces is normal with mean μ = 150 Ω and standard deviation σ = 5 Ω. What percentage of the resistors will have resistance between and Between 148 Ω and 152 Ω? Between 140 Ω and 160 Ω?
8. The breaking strength X [kg] of a certain type of plastic block is normally distributed with a mean of 1500 kg and a standard deviation of 50 kg. What is the maximum load such that we can expect no more than 5% of the blocks to break?
9. If the mathematics scores of the SAT college entrance exams are normal with mean 480 and standard deviation 100 (these are about the actual values over the past years) and if some college sets 500 as the minimum score for new students, what percent of students would not reach that score?
10. A producer sells electric bulbs in cartons of 1000 bulbs. Using (11), find the probability that any given carton contains not more than 1% defective bulbs, assuming the production process to be a Bernoulli experiment with p = 1%(= probability that any given bulb will be defective). First guess. Then calculate.
11. If sick-leave time X used by employees of a company in one month is (very roughly) normal with mean 1000 hours and standard deviation 100 hours, how much time t should be budgeted for sick leave during the next month if t is to be exceeded with probability of only 20%?
12. If the monthly machine repair and maintenance cost X in a certain factory is known to be normal with mean $12,000 and standard deviation $2000, what is the probability that the repair cost for the next month will exceed the budgeted amount of $15,000?
13. If the resistance X of certain wires in an electrical network is normal with mean 0.01 Ω and standard deviation 0.001 Ω, how many of 1000 wires will meet the specification that they have resistance between 0.009 and 0.011 Ω?
14. TEAM PROJECT. Normal Distribution. (a) Derive the formulas in (6) and (7) from the appropriate normal table.

    (b) Show that Φ(−z) = 1 − Φ(z). Give an example.

    (c) Find the points of inflection of the curve of (1).

    (d) Considering Φ²(∞) and introducing polar coordinates in the double integral (a standard trick worth remembering), prove

    

    (e) Show that σ in (1) is indeed the standard deviation of the normal distribution. [Use (12).]

    (f) Bernoulli's law of large numbers. In an experiment let an event A have probability P(0 < p < 1), and let X be the number of times A happens in n independent trials. Show that for any given ∈ > 0,

    

    (g) Transformation. If X is normal with mean μ and variance σ², show that X* = c₁X + c₂(c₁ > 0) is normal with mean μ* = c₁μ + c₂ and variance .

15. WRITING PROJECT. Use of Tables, Use of CAS. Give a systematic discussion of the use of Tables A7 and A8 for obtaining P(X < b), P(X > a), P(a < X < b), P(X < c) = k, P(X > c) = k, as well as P(μ − c < X < μ + c) = k; include simple examples. If you have a CAS, describe to what extent it makes the use of those tables superfluous; give examples.

EXAMPLE 1 Two-Dimensional Discrete Distribution

If we simultaneously toss a dime and a nickel and consider

then X and Y can have the values 0 or 1, and the probability function is

EXAMPLE 2 Two-Dimensional Uniform Distribution in a Rectangle

Let R be the rectangle α₁ < x β₁, α₂ < y β₂. The density (see Fig. 524)

defines the so-called uniform distribution in the rectangle R; here k = (β₁ − α₁)(β₂ − α₂) is the area of R. The distribution function is shown in Fig. 525.

EXAMPLE 3 Marginal Distributions of a Discrete Two-Dimensional Random Variable

In drawing 3 cards with replacement from a bridge deck let us consider

The deck has 52 cards. These include 4 queens, 4 kings, and 4 aces. Hence in a single trial a queen has probability and a king or ace . This gives the probability function of (X, Y),

and f(x, y) = 0 otherwise. Table 24.1 shows in the center the values of f(x, y) and on the right and lower margins the values of the probability functions f₁(x) and f₂(y) of the marginal distributions of X and Y, respectively.

EXAMPLE 4 Independence and Dependence

In tossing a dime and a nickel, X = Number of heads on the dime, Y = Number of heads on the nickel may assume the values 0 or 1 and are independent. The random variables in Table 24.1 are dependent.

THEOREM 1 Addition of Means

The mean (expectation) of a sum of random variables equals the sum of the means (expectations) , that is,

THEROEM 2 Multiplication of Means

The mean (expectation) of the product of independent random variables equals the product of the means (expectations) , that is,

PROOF

If X and Y are independent random variables (both discrete or both continuous), then E(XY) = E(X)E(Y). In fact, in the discrete case we have

and in the continuous case the proof of the relation is similar. Extension to n independent random variables gives (26), and Theorem 2 is proved.

THEOREM 3 Addition of Variances

The variance of the sum of independent random variables equals the sum of the variances of these variables.

PROBLEM SET 24.9

1. Let f(x, y) = k when 8 x 12 and 0 y 2 and zero elsewhere. Find k. Find P(X 11, 1 Y 1.5) and p(9 X 13, Y 1).
2. Find P(X > 4, Y > 4) and P(X 1, Y 1) if (X, Y) has the density if x 0, y 0, x + y 8.
3. Let f(x, y) = k if x > 0, y > 0, x + y < 3 and 0 otherwise. Find k. Sketch f(x, y). Find P(X + Y 1), P(Y > X).
4. Find the density of the marginal distribution of X in Prob. 2.
5. Find the density of the marginal distribution of Y in Fig. 524.
6. If certain sheets of wrapping paper have a mean weight of 10 g each, with a standard deviation of 0.05 g, what are the mean weight and standard deviation of a pack of 10,000 sheets?
7. What are the mean thickness and the standard deviation of transformer cores each consisting of 50 layers of sheet metal and 49 insulating paper layers if the metal sheets have mean thickness 0.5 mm each with a standard deviation of 0.05 mm and the paper layers have mean 0.05 mm each with a standard deviation of 0.02 mm?
8. Let X [cm] and Y [cm] be the diameters of a pin and hole, respectively. Suppose that (X, Y) has the density

   

   and 0 otherwise. (a) Find the marginal distributions. (b) What is the probability that a pin chosen at random will fit a hole whose diameter is 1.00?

9. Using Theorems 1 and 3, obtain the formulas for the mean and the variance of the binomial distribution.
10. Using Theorem 1, obtain the formula for the mean of the hypergeometric distribution. Can you use Theorem 3 to obtain the variance of that distribution?
11. A 5-gear assembly is put together with spacers between the gears. The mean thickness of the gears is 5.020 cm with a standard deviation of 0.003 cm. The mean thickness of the spacers is 0.040 cm with a standard deviation of 0.002 cm. Find the mean and standard deviation of the assembled units consisting of 5 randomly selected gears and 4 randomly selected spacers.
12. If the mean weight of certain (empty) containers is 5 lb the standard deviation is 0.2 lb, and if the filling of the containers has mean weight 100 lb and standard deviation 0.5 lb, what are the mean weight and the standard deviation of filled containers?
13. Find P(X > Y) when (X, Y) has the density

    

    and 0 otherwise.

14. An electronic device consists of two components. Let X and Y [years] be the times to failure of the first and second components, respectively. Assume that (X, Y) has the density f(x, y) = 4e^−2(x+y) if x > 0 and y > 0 and 0 otherwise. (a) Are X and Y dependent or independent? (b) Find the densities of the marginal distributions. (c) What is the probability that the first component will have a lifetime of 2 years or longer?
15. Give an example of two different discrete distributions that have the same marginal distributions.
16. Prove (2).
17. Let (X, Y) have the probability function

    

    Are X and Y independent?

18. Let (X, Y) have the density

    

    and 0 otherwise. Determine k. Find the densities of the marginal distributions. Find the probability

    

19. Show that the random variables with the densities

    

    and

    

    if 0 x 1, 0 y 1 and f(x, y) = 0 and g(x, y) = 0 elsewhere, have the same marginal distribution.

20. Prove the statement involving (18).