The exponential distribution models the waiting time between two events in a Poisson process. A Poisson process is a process that follows a Poisson distribution in which events occur unpredictably with a known average rate. The exponential distribution can be described as the continuous limit of the geometric distribution, and is also Markovian (memoryless).

A memoryless random variable exhibits the property that its future state depends only on relevant information about the current time and not the information from further in the past. An example of modeling a Markovian/memoryless random variable is modeling short-term stock price behavior based on the idea that it follows a random walk. This leads to what is called the efficient market hypothesis in finance. For more information, refer to http://en.wikipedia.org/wiki/Random_walk_hypothesis.

The PDF of the exponential distribution is given by f(x)= λe^-λx. The expectation and variance are given respectively by the following expressions:

For reference, refer to the link at http://en.wikipedia.org/wiki/Exponential_distribution.

The plot of the distribution and code is given as follows:

    In [15]: import scipy.stats
             clrs = colors.cnames
               x = np.linspace(0,4, 100)
            expo = scipy.stats.expon
            lambda_ = [0.5, 1, 2, 5]
           plt.figure(figsize=(12,4))
           for l,c in zip(lambda_,clrs):
               plt.plot(x, expo.pdf(x, scale=1./l), lw=2,
                          color=c, label = "$lambda = %.1f$"%l)
                 plt.legend()
                 plt.ylabel("PDF at $x$")
          plt.xlabel("$x$")
          plt.title("Pdf of an Exponential random variable for various $lambda$");
  

The following is the output: