The moment-generating function of a continuous random variable x with a p.d.f. f(x), if it exists, is defined by: M(t) = int_{-infty}^{infty} e^{tx}f(x)dx, for h > 0, dots, y alpha heta u = frac{(1 - heta t)x}{ heta} 1 - heta t > 0 - h and some h > 0, dots, y
Use the definition in question number 3 above and find the moment generating function of the gamma distribution with parameters alpha and heta. Hint: Use change variables: u = frac{(1 - heta t)x}{ heta}, where 1 - heta t > 0. [10 points]
The moment-generating function of a continuous random variable X with a p.d.f. f(x) if it exists, is defined by: M(t) = e^{f(x)dx}, for -h < t < h and some h > 0. y
Use the definition in question number 3 above and find the moment generating function of the gamma distribution with parameters a and . Hint: Use change variables: 1 - Otx where 1 - t > 0. [10 points]