Paper 2, Section II, F

Probability | Part IA, 2015

Consider the function

ϕ(x)=12πex2/2,xR\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad x \in \mathbb{R}

Show that ϕ\phi defines a probability density function. If a random variable XX has probability density function ϕ\phi, find the moment generating function of XX, and find all moments E[Xk]E\left[X^{k}\right], kNk \in \mathbb{N}.

Now define


Show that for every x>0x>0,


Typos? Please submit corrections to this page on GitHub.