Question

Having troubles with question 2. Please help

2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters α2 and β, and that X1 and X2 are independent. show that the variable Y = X1 + X2 has a Gamma distribution with parameters α (e1 WherCDa to show that the sum of two independent chi-square random (c) (d) (e) Use (b) to show that the sum of two independent chi-square random variables is a chi-square random variable. Suppose X has a chi-square distribution with n df. Use the derivatives of the mgf of X to show that E[X] n and V[X] 2n Suppose that Y has an exponential distribution with mean B. Use the derivatives of the mgf of γ to show that E[Yn] = n! β". for every positive integer n

Answer 1

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