A family of random variables has MGF ψ(t) = eat2 , t ∈ R, where the value of a is a parameter defining the distribution. (a)  Find the mean and variance of X as a function of a. (b)  Assume X1 , . . . , Xn , a random sample of size n, is drawn from the distribution for a given value of a. Find the MGF, mean and variance of the sum Y = ∑ni=1Xi. (c)  Let Z = Y/n = 1/n ∑ni=1Xi be the sample mean of this random sample. Find the MGF, mean, and variance of Z.

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