If X has the normal distribution with mean 0 and variance 1, then MX (t) = Z et x 1 2 e1 2

If X has the normal distribution with mean 0 and variance 1, then MX (t) =

Z

∞

−∞

et x 1

√2π

e−1 2 x2 dx

= e 1

2 t 2 Z

∞

−∞

1

√2π

e−1 2 (x−t)2 dx

= e 1

2 t 2

, (7.44)