For any real number r 0 the gamma function of r is given by

For any real number r > 0, the gamma function (of r) is given by Gamma (r) = integral_0^infinity x^r - 1 e^-x dx. A property of Gamma (r) is the following: Gamma (r + 1) = r Gamma (r). Let X be a random variable with pdf f(x) = lambda^r / Gamma (r) x^r - 1 e^-lambda x, with x > 0 and parameters lambda > 0, r > 0. Show that E(X) = r/lambda. Show that Var(X) = r/lambda^2.

Solution