In the following, we will see how to use the Exponential, Gamma, and Normal distributions to prove a very useful and famous approximation called Stirling's approximation for calculating large factorials.
Suppose X and Y are independent observations from a Gamma(1, ) and a Gamma(2, ) distribution, respectively. Show that Z = X + Y is distributed as a Gamma(a + a, ) distribution.
(a) Show that the distribution of X + Y can be calculated by evaluating the integral: f(x+y)(z) = 2^(x-1)e^(-Xx)(z-x)^(a-1)e^(-x)(-x) f(x)f(y(z-x))dx = dx. r(a) T(a^2) (4)
(b) Show that the answer above simplifies to 1 + a^2e^(-Xz) f(x+y)(z) = x^(P-2)(x-2-1X) r(a)I(a^2)Jo (4)
(c) Use the change of variables z = t to show that Z ~ Gamma(a + 2, A) - I(aT(a^2) Hint: 1/(a1+a2) (4)
(d) Show that an Exponential distribution is a special case of a Gamma distribution and so a sum of n independent X; Exponential() distributions is a Gamma(n, A) distribution. Hint: T(n) = (n-1)! for n = 0, 1, 2,... (4)
(e) Suppose you have n independent and identically distributed X; Exponential = 1 random variables. Show that for the variable n+2/n f(s)(s)ds where S ~ Gamma(n, 1) distribution Vn (4)
(f) As n -> infinity, what is an alternative way of approximating F(z) = P(Z < z) (4)
(g) A version of the Fundamental Theorem of Calculus says that
dz Jc
(n + z/n)^(n-1) I(n)
V2T
(4)
