Let , be positive real numbers with > 2 and Z Gam (, ), with Random variable Y = 1/Z is said

Let α, β be positive real numbers with α > 2 and Z ∼ Gam (α, β), with

Random variable Y = 1/Z is said to follow an inverse gamma distribution, denoted by Y ∼ IGam(α, β).

(a) Compute E[Z] and E
Z−1
.

(b) Derive fY (y; α, β), the p.d.f. of Y , and interpret parameter β.

(c) Derive a useful expression for the c.d.f. of Y .

(d) Derive a useful expression for µ
r, the rth raw moment of Y . Simplify for E [Y ] and V (Y ).

(e) When are E [Z] and E
Z−1
reciprocals of one another, i.e. what conditions on α and β are necessary so that E
1/Z
= 1/E [Z]?

fz (z; , )= exp(-Bz) (0.00) (2). ()