3.3.24. Let X₁, X2 be two independent random variables having gamma distribu- tions with parameters a₁ = 3, 3₁ = 3 and 02 = 5, 3₂= 1, respectively. (a) Find the mgf of Y = 2X₁ +6X₂. (b) What is the distribution of Y?

12. i. State Central limit theorem and tell what is the main theme of it. ii. Show that if the random variables (X, Y) follow the probability density of a bivariate normal distribution and if the correlation p = 0, X and Y are independent. (Show that f(x, y) = g(x) h(y))

Please show step by step explanations. Thank you.

Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)