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$$f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$$

where $0.

So these are three terms all multiplied to give you an ugly distribution function where $\alpha>0$ is some parameter, $\beta>0$ is a parameter. $\Gamma$ refers to the Gamma function.

This very closely resembles the Gamma Distribution function but not quite and I don't know how to find the expectation and variance for $X$ with the given distribution function.

I tried to go the route of finding the moment generating function to make the distribution resemble a gamma and use the fact that the density would integrate to one but the $(1-\beta x)$ term really complicates things. Not sure what to do.

Help.

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