Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 4F / b

Table of contents
- Solution b

Solution

Completing the square in

x_{1}

in the joint density shows that the conditional distribution of a bivariate normal variable is

X_{1} ∣ X_{2} = x_{2} \sim N (μ_{1} + ρ \frac{σ _{1}}{σ _{2}} (x_{2} - μ_{2}), σ_{1}^{2} (1 - ρ^{2})) .

Thus its conditional density is

\frac{1}{2 π σ _{1}^{2} ( 1 - ρ ^{2} )} exp - \frac{( x _{1} - μ _{1} - ρ \frac{σ _{1}}{σ _{2}} ( x _{2} - μ _{2} ) ) ^{2}}{2 σ _{1}^{2} ( 1 - ρ ^{2} )} .

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