Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 12F / a / Solution

The nonsingular bivariate normal distribution has ${\sigma }_1,{\sigma }_2>0$ and $|\rho |<1$. Put $a=(x_1-{\mu }_1)/{\sigma }_1$ and $b=(x_2-{\mu }_2)/{\sigma }_2$. Completing the square givesConsequently the joint probability density function factors asFor fixed $x_1$, the second factor is a normalized normal distribution density in $x_2$, with mean ${\mu }_2+\rho {\sigma }_{2}a$ and standard deviation ${\sigma }_2\sqrt{1-{\rho }^2}$. Its integral is one; explicitly substitute $w=(b-\rho a)/\sqrt{1-{\rho }^2}$ in the Gaussian integral. Thus the marginal density is