 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/28k/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Conditionally on

Y_{1}, \dots, Y_{m}

X^{*}

is the weighted empirical distribution

F_{m}^{*} (t) = P (X^{*} \leq t ∣ Y_{1}, \dots, Y_{m}) = \frac{m ^{- 1} \sum _{i = 1}^{m} w ( Y _{i} ) 1 _{{Y_{i} \leq t}}}{m ^{- 1} \sum _{i = 1}^{m} w ( Y _{i} )}, w (y) = \frac{f ( y )}{h ( y )} .

For fixed

t

\frac{1}{m} \sum_{i = 1}^{m} w (Y_{i}) 1_{{Y_{i} \leq t}} ⟶ \int_{(- \infty, t]} f (y) d y = F (t),

while

\frac{1}{m} \sum_{i = 1}^{m} w (Y_{i}) ⟶ \int f (y) d y = 1.

The ratio therefore converges almost surely to

F (t)

 New to topics? Read the docs here!