4. Let X have probability density function f_X(x) := (2^2)^(1/2)exp[(x)^2/(2^2)], where is a real number and ^2 is a positive real number. This is a normal distribution. Assume that E[X] =,Var[X] =^2, and MX(t) :=E[exp(tX)] =exp[t+^2t^2/2].

a. Find the probability density function of Y:= exp[X]. This is a lognormal distribution.

b. Find E[Y],E[Y^2], and then Var[Y].

c. Let a be a positive real number. Suppose X is a nonnegative random variable, and put Y:=X/(X+a). Show that Y has moments of all orders