Let X be a random variable with probability density function f(x), and let M(t) =E[e] be its moment generating function The tilted density function

f, is defined by f(x) = e"f(x) M(t) Let X, have density function f

(a) Show that for any function h(x) = E[h(X)] M(t)E[exp{-tx,}h(X)]

(b) Show that, for > 0, P{X> a} M(t)eP{X, > a}.

(c) Show that if P{X* > a} = 1/2 then min M(t)e M(*)e". Lightchat