Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 1 / 12F / a

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- Solution a

Solution

The standard normal density is

ϕ (z) = \frac{1}{2 π} e^{- z^{2} /2}, z \in R .

It is nonnegative, and the Gaussian integral gives

\int_{- \infty}^{\infty} ϕ (z) d z = 1

, so it is a probability density function. Completing the square gives the moment-generating function of a standard normal variable

M_{Z} (θ) = \int e^{θ z} ϕ (z) d z = e^{θ^{2} /2} .

Thus

E Z = M_{Z}^{'} (0) = 0

and

var (Z) = M_{Z}^{''} (0) - M_{Z}^{'} (0)^{2} = 1.

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