Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 1 / a

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- Solution a

Solution

A spherical shell contributes a factor

4 π r^{2}

to the number of stars. Since

\int_{0}^{\infty} r^{2} e^{- r / r_{0}} d r = 2 r_{0}^{3}

, the normalized distance density is

p (r_{s}) = \frac{r _{s}^{2}}{2 r _{0}^{3}} e^{- r_{s} / r_{0}}, r_{s} > 0.

Thus

C = (2 r_{0}^{3})^{- 1}

,

α = 2

, and

β = 1

. This is a gamma distribution with shape three and scale

r_{0}

.

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