Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 18H / d

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- Solution d

Solution

(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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