=+16. Suppose X has gamma density x1ex/() on (0, ), where is not necessarily an integer.

=+16. Suppose X has gamma density λβxβ−1e−λx/Γ(β) on (0, ∞), where β

is not necessarily an integer. Show that X has characteristic function

( λ

λ−it )β and Laplace transform ( λ

λ+t )β. Use either of these to calculate the mean and variance of X. (Hint: For the characteristic function, derive and solve a differential equation. Alternatively, calculate the Laplace transform directly by integration and show that it can be extended to an analytic function in a certain region of the complex plane.)