 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-1/18h/d/solution

Codex  0 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.

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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