since the integrand in the latter integral is the density function of the normal distribution with mean t and variance 1, and thus has integral 1. The moment generating function MX (t) exists for all t ∈ R. △

Example 7.45 If X has the exponential distribution with parameter λ, then MX (t) =

Z

∞

0 et xλe−λx dx =





λ

λ − t if t < λ,

∞ if t ≥ λ,

(7.46)

so that MX (t) exists only for values of t satisfying t < λ. △

Example 7.47 If X has the Cauchy distribution, then MX (t) =

Z

∞

−∞

et x 1

π(1 + x2)

dx =

(

1 if t = 0,

∞ if t 6= 0, so that MX (t) exists only at t = 0.