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 pdf of theta
Consider parameters $\displaystyle \displaystyle k, \theta $ such that $\displaystyle \displaystyle \mu = k\theta $ and $\displaystyle \displaystyle \sigma^2 =k\theta^2$
You are told $\displaystyle \displaystyle k\theta =10 , k\theta^2 = 5$ solve for $\displaystyle \displaystyle k, \theta$
The pdf will then be $\displaystyle \displaystyle P(x) =\frac{x^{k-1}e^{\frac{-x}{\theta}}}{\Gamma (k)\theta^k}$