Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 4 / a

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- Solution a

Solution

For

0 < p < 1

, the Bernoulli distribution has mass

Pr (Y = y) = p^{y} (1 - p)^{1 - y} = exp [y lo g \frac{p}{1 - p} + lo g (1 - p)], y \in {0, 1} .

In exponential dispersion family notation

exp {[y θ - b (θ)] / ϕ + c (y, ϕ)}

, take

θ = lo g \frac{p}{1 - p}, b (θ) = lo g (1 + e^{θ}), ϕ = 1, c (y, 1) = 0.

The support is

{0, 1}

; outside it the probability is zero. The natural parameter of an exponential family is the log odds, and the dispersion parameter is fixed rather than freely estimated. Differentiating the cumulant gives

E Y = b^{'} (θ) = \frac{e ^{θ}}{1 + e ^{θ}} = p, Var (Y) = ϕ b^{''} (θ) = \frac{e ^{θ}}{( 1 + e ^{θ} ) ^{2}} = p (1 - p) .

At

p = 0

or

1

the degenerate distributions are limiting boundary cases with natural parameter

- \infty

or

+ \infty

.

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