Random variable X has a generalized exponential distribution, denoted X GED (p), if X has p.d.f. Let Y = X for >

Random variable X has a generalized exponential distribution, denoted X ∼

GED (p), if X has p.d.f.

Let Y = σ X for σ > 0, i.e. σ is a scale parameter.

(a) Verify that fX (x; p) integrates to one.

(b) Derive an expression for the c.d.f. of X in terms of the incomplete gamma function.

(c) Give the density of Y and parameter values p and σ such that the Laplace and normal distributions arise as special cases.

(d) Compute V (Y ) and verify that it agrees with the special cases of Laplace and normal.

(e) Show that, if X ∼ GED (p), then Y = |X|
p ∼ Gam (1/p). This fact can be used for simulating GED r.v.s (which is not common to most software), via simulation of gamma r.v.s (for which most statistical programming software packages do have routines). In particular, generate Y ∼ Gam (1/p), set X = Y 1/p, and then attach a random sign to X, i.e. with probability 1/2, multiply X by negative one.

P 2 (p-1) exp(-1x1), pe R0. fx (x; p) = 2r (p-1)