Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 1 / d

Table of contents
- Solution d

Solution

For an intrinsic probability density function

P (q)

, the law of total probability averages the conditional density from part c. An object observed with ratio

Q

can only have

q \leq Q

, so

P (Q) = Q \int_{0}^{Q} \frac{P ( q )}{1 - q ^{2} Q ^{2} - q ^{2}} d q, 0 < Q < 1 .

This is an Abel transform of

P (q) / 1 - q^{2}

. Its normalization follows by reversing the order of integration and using

\int_{q}^{1} P (Q ∣ q) d Q = 1

.

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