Show that we can assume beta 1 by using the transformation

Question

We saw in Example 2.2 that, if α ∈ N, the gamma distribution Ga(α, β) can be represented as the sum of α exponential random variables ?_i ∼ Exp(β). When α ∉ N, this representation does not hold.

a. Show that we can assume β = 1 by using the transformation y = βx.

b. When the G(n, 1) distribution is generated from an Exp(λ) distribution, determine the optimal value of λ.

c. When α ≥ 1, show that we can use the Accept-Reject algorithm with candidate distribution Ga(a, b) to generate a Ga(α, 1) distribution, as long as a ≤ α. Show that the ratio f /g is b^-ax^α-a exp{-(1 - b)x}, up to a normalizing constant, yielding the bound

M = b^-a((α-a)/(1-b)e)^α-a

for b

d. Show that the maximum of b^-a(1 - b)^α-a is attained at b = a/α, and hence the optimal choice of b for simulating Ga(α, 1) is b = a/α, which gives the same mean for Ga(α, 1) and Ga(a, b).

e. Defend the choice of a =[α] as the best choice of a among the integers.