Question

The Exponential distribution characterizes the behavior of a random variable that describes the time between events that occur continuously and at a constant average rate λ. The CDF for the Exponential distribution is given by F(y) = 1 − e ^−λy .

(a) Find the PDF of the Exponential distribution.

(b) Intuitively, do you think the expected value of the Exponential distribution depends positively or negatively on λ. Explain.

(c) Prove that E[Y ] = 1 λ .

(d) Consider an unemployed individual looking for work who knows that job offers arrive at an average rate of 2 per month. Assuming all job offers are accepted, for how long should she expect to remain unemployed?

(e) Suggest a real-life process that could potentially be described by an Exponential distribution.

Answer 1

a) Differentiate with respect to y gives the PDF as

y>0

b)  The expected value of the Exponential distribution depends negatively on , Since as increases, decreases

c) By definition of expectation we have

d) Using c) we have the expected number of months of unemployment is

e) Pure Birth process is an example of Exponential distribution.