A random variable is a variable whose value is unknown, or a variable for which the value changes over different iterations of the experiment.

For example, when we roll a die, the outcome of rolling the dice will vary over different iterations and hence the outcome becomes a random variable.

A random variable is discrete if the outcome of the random variable is limited to a few possible outcomes.

For example, the outcome of rolling a fair dice can only be 1, 2, 3, 4, 5, or 6; it cannot be a number beyond that. Thus, the outcome is limited to only a few possible values.

In the previous example, whenever a die is rolled, there is a probability associated with each outcome. For example, if a fair dice is rolled once, the probability that the outcome is 4 is 1/6, as all outcomes have an equal chance of being obtained.

Given that the pmf assigns a probability number to each possible outcome of the experiment, we can think of it as the pmf assigning a mass (weight) to each possible outcome, where the mass is high if the likelihood of occurrence (probability) is high.

There can be multiple scenarios to specify a pmf:

• Binomial discrete distribution: Calculating the probability of the outcome when there are only two possible outcomes
• Multivariate discrete distribution: Calculating the probability when there are multiple possible outcomes