2.15 The random variable whose probability density function is given by f(x) = 1 2ex , if x 0 1 2ex , if x

2.15 The random variable whose probability density function is given by f(x) = 1 2λeλx , if x ≤ 0 1

2λe−λx , if x > 0, is said to have a Laplace, sometimes called a double exponential, distribution.

a) Verify that the density above defines a proper probability distribution.

b) Find the distribution function F(x) for a Laplace random variable.

Now, let X and Y be independent exponential random variables with parameter λ. Let I be independent of X and Y and equally likely to be 1 or

−1.