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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2010

Paper 2

Now complete the square in the other order:
$$
a^2-2\rho ab+b^2=(a-\rho b)^2+(1-\rho^2)b^2.
$$
Divide the joint <probability density function> by the positive marginal density
$$
f_{X_2}(x_2)=\frac1{\sqrt{2\pi}\sigma_2}e^{-b^2/2}.
$$
The factor $e^{-b^2/2}$ cancels, giving the <conditional probability density>
$$
f_{X_1\mid X_2=x_2}(x_1)=
\frac1{\sqrt{2\pi}\sigma_1\sqrt{1-\rho^2}}
\exp\left[-\frac{(a-\rho b)^2}{2(1-\rho^2)}\right].
$$
Since
$$
a-\rho b=\frac{x_1-\left(\mu_1+\rho\sigma_1(x_2-\mu_2)/\sigma_2\right)}{\sigma_1},
$$
this is a normalized <normal distribution> density. \b[Hence]
$$
\boxed{X_1\mid X_2=x_2\sim
N\left(\mu_1+\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2),
\ \sigma_1^2(1-\rho^2)\right).}
$$
Conditioning on an exact observation is interpreted through this <conditional probability density>, rather than conditioning by a ratio of event <probabilities>.

Solution

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

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

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