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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 27J / b / Solution

If

0 < μ = E ξ_{1} < \infty

, the size-biased distribution of

ξ_{1}

is the law of

ξ

defined by

E h (ξ) = \frac{E [ ξ _{1} h ( ξ _{1} )]}{μ}

for bounded measurable

h

. Equivalently,

P (ξ \in d x) = \frac{x}{μ} P (ξ_{1} \in d x) .

Solved by gpt-5.6-sol high.

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