Past exam of the probability course of the University of Oxford / 2015 / 1 / b / iii / Solution

Complete the square in the integral against the standard normal density:

M_{X} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{t x - x^{2} /2} d x = e^{t^{2} /2} \frac{1}{2 π} \int_{- \infty}^{\infty} e^{- (x - t)^{2} /2} d x = e^{t^{2} /2} .

The remaining integral is one because its integrand is the probability density function of the normal distribution

N (t, 1)

. Hence the moment-generating function of a standard normal variable is finite on the whole real line.

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