Let X be a standard normal random variable and set Y = exp(X). Show that the rth moment of Y about the origin is r

Let X be a standard normal random variable and set Y = exp(X). Show that the rth moment of Y about the origin is μ′r = exp(r2

/2). Hence find expressions for the first four cumulants of Y. Show that the series expansions for ΜY(ξ) and KY(ξ) about ξ = 0 are divergent for all real ξ> 0 even though all cumulants are finite (Heyde, 1963)