Q

a. The random variable X is Poisson distributed with expected value 7t. Verify that E[Ag(X + 1) g(X)] = 0 for any bounded function g(x) on the integers 0,1,.... b. Let X be a random variable on the integers 0, 1, and A > 0 a given number. Prove that X has a Poisson distribution with expected value A if E[}\\g(X + 1)Xg(X)] = 0 for any bounded function g(x) on the integers 0,1,.... Hint: take the function g(x) dened by g(r) = 1 and g(x) = 0 for x r with r a xed nonnegative integer. c. The density function of the continuous random variable X is given by f(x) = C(x + W) for O