Hence, using the tower property, determine ELXX]. You may use standard properties of the Gaussian distribution without proof. (d) Suppose that (Y;&21) is a sequence of n independent random variables each having a continuous distribution with density ka fy(y)= (1+ y/2)+1/2 VER where ka is a constant depending on a. Prove that, for all r > 2 P(Y2) - 2a and hence calculate P(Y, 2 n/(20) for infinitely many n). (e) Suppose that the sequence X1, X2, X, is extended to an infinite sequence of random variables (X;k 2 1), independent given T, and with each X, satisfying equation (1). How does the behaviour of the sequence (Y; & 2 1) compare with the behaviour of the sequence (X; k 1)? You may quote results from the lectures on the behaviour of a sequence of independent random variables each having a Gaussian distribution. [5] [3] +