Question

"Let X and Y be independent random variables distributed as exponential with parameters λ and μ, respectively. Define W = X - Y and Z as follows: Z = (X - Y) if I = 1, and Z = 0 if I = 0, where I is a Bernoulli random variable with success probability P(I=1) = (λ + μ) / μ. Show, using moment generating functions, that W and Z have the same distribution.

Answer 1

To show that W and Z have the same distribution, we need to show that their moment generating functions are equal.

Step-by-step explanation:

To show that W and Z have the same distribution, we need to show that their moment generating functions are equal. The moment generating function of W can be found by taking the moment generating function of X and Y and substituting them into the moment generating function of W = X - Y. The moment generating function of Z can be found by taking the moment generating function of X - Y and multiplying it by P(I=1) and adding it to the moment generating function of 0 multiplied by P(I=0). By comparing the two moment generating functions, we can see that they are equal.