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 Incoming links: Conditional distribution of a bivariate normal variable

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 18H / d / Solution Created 2026-09-24 Updated 2026-10-03

(X_{1}, T)

is jointly normal, with

E X_{1} = μ, E T = 2 μ, var (X_{1}) = 1, var (T) = 2, cov (X_{1}, T) = 1.

The conditional distribution of a bivariate normal variable is therefore

X_{1} ∣ T = t \sim N (\frac{t}{2}, \frac{1}{2}) .

Equivalently, its conditional density is

f_{X_{1} ∣ T = t} (x) = \frac{1}{π} exp [- (x - \frac{t}{2})^{2}] .

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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 4F / b / Solution Created 2026-09-24 Updated 2026-10-03

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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