Question
Suppose $X$ is a gamma random variable with parameters $\beta$ and $\alpha$. Show directly from the definition that$$X^*[\theta]=\left(\frac{\alpha}{\theta+\alpha}\right)^\beta$$
Step 1
We are given that $X$ is a gamma random variable with parameters $\beta$ and $\alpha$. The probability density function (pdf) of a gamma random variable $X$ with shape parameter $\beta$ and rate parameter $\alpha$ is given by: $$ f_X(x) = Show more…
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