# Question

Prove the following properties of the Gamma function.

(a) Γ (n) = (n–1)! For n = 1,2,3,….

(b) Γ (x + 1) = x Γ (x)

(c) Γ (1 / 2) = √x.

(a) Γ (n) = (n–1)! For n = 1,2,3,….

(b) Γ (x + 1) = x Γ (x)

(c) Γ (1 / 2) = √x.

## Answer to relevant Questions

Prove the following properties of conditional CDFs. (a) (b) (c) (d) Suppose V is a uniform random variable, (a) Find the conditional PDF, fv|{v > 1}(v) . (b) Find the conditional PDF, fv|{1/2 < v < 3/2}(v). (c) Find the conditional CDF,fv|{1/2 < v < 3/2}(v) Suppose we are given samples of the CDF of a random variable. That is, we are given Fn = Fx (xn) at several points, xn Ɛ { x1, x2, x3,….xk. After examining a plot of the samples of the CDF, we determine that it appears ...Suppose X is a Gaussian random variable with mean µ X and variance σ2x . Suppose we form a new random variable according to Y= aX+ b for constants a and b. (a) Prove that Y is also Gaussian for any a ≠ 0. (b) What ...A Cauchy random variable has a PDF (a) Find the characteristic function, ϕX(ω) . (b) Show that the derivatives dk / dωk (ϕX(ω)) do not exist at ω = 0.What does this mean?Post your question

0