2.1. Use the probability distribution function p(z) = exp(z2/2)/ 2 for N(0,1) random variables Z and integration by parts to deduce that E[Zp] =

2.1. Use the probability distribution function p(z) = exp(−z2/2)/

√

2π for N(0,1) random variables Z and integration by parts to deduce that E[Zp] = (p−1)(p−3) . . .1 for even integers p. Hence deduce forWiener processesW(t) that E[W(t)p] = (p−

1)(p−3) . . .1· tp/2 (for even integers p) by writing, at any given time, W = Z

√

t where Z ∼ N(0,1) .