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Chapter 1: Q7.2-4E (page 1)

Suppose that the prior distribution of some parameter \(\theta \) is a gamma distribution for which the mean is 10 and the variance is 5. Determine the prior p.d.f. of \(\theta \).

Short Answer

Expert verified

Prior pdf of is \(\xi \left( \theta \right) = \frac{{{2{20}}}}{{\left| \!{\overline {\, {\left( {20} \right)} \,}} \right. }}{\theta {19}}\exp \left( { - 2\theta } \right)\)

Step by step solution

01

Given information

Let the prior distribution of some parameter \(\theta \) follows gamma distribution with mean 10 and the variance 5.

02

Calculating prior pdf of \(\theta \)

Let if\(\alpha \)and\(\beta \)be the parameters of gamma distribution, then we must have

Mean :\(\frac{\alpha }{\beta }\)

That is\(\frac{\alpha }{\beta } = 10\)

variance :\(\frac{\alpha }{{{\beta ^2}}}\)

that is\(\frac{\alpha }{{{\beta ^2}}} = 5\)

Therefore\(\alpha = 20\)and\(\beta = 2\)

Hence the prior pdf of\(\theta \)is as follows

For \(\theta > 0:\)

\(\xi \left( \theta \right) = \frac{{{2^{20}}}}{{\left| \!{\overline {\, {\left( {20} \right)} \,}} \right. }}{\theta ^{19}}\exp \left( { - 2\theta } \right)\)

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