 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2011

Paper 2

Section I

The <moment-generating function> is $M_Y(t)=\mathbb E[e^{tY}]$ for real $t$ where this expectation is finite; a finite neighborhood of zero is usually required when speaking of an existing <moment-generating function>. For a <standard normal distribution>, completing the square gives
$$
M_Y(t)=\frac1{\sqrt{2\pi}}\int_{\mathbb R}e^{ty-y^2/2}\,dy
=e^{t^2/2}\frac1{\sqrt{2\pi}}\int_{\mathbb R}e^{-(y-t)^2/2}\,dy.
$$
The translated <Gaussian density> integrates to one, so
$$
\boxed{M_Y(t)=e^{t^2/2}\quad(t\in\mathbb R).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 3F / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 3F / b / Solution

 Ancestors (12)

 Discussion (0) New discussion

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 3F / b / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion