Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 44 / 3 / a

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- Solution a

Solution

Write the new measurement as

Y_{i 1} = θ_{i} + ε_{i 1}

, with independent

θ_{i} \sim N (μ, τ^{2})

and

ε_{i 1} \sim N (0, σ_{i}^{2})

. Adding independent normal distributions gives the prior predictive distribution

Y_{i 1} \sim N (μ, τ^{2} + σ_{i}^{2}) .

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