Finding the Moment Generating Function of a Transformed Random Variable

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Suppose I have a random variable $X$ governed by the Gamma distribution.

If I apply a transformation like $Y = \ln\bigg( \frac{X}{1+X}\bigg)$, could I use properties of moment generating functions to find $M_{Y}(t)$ instead of performing $\int e^{ty}f(y) \mathrm{dy}$?

By properties I mean use something like $M_{\ln(X/(1-X))(t)} = E[e^{\ln(X/(1-X))}]$

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1 Answer

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Suppose that $g(x)=\ln(\frac{x}{1+x})$. Law of the unconscious statician states that

$$\mathbb{E}[h(X)]=\int h(x)f_X(x)\,dx$$Thus, $$M_Y(t)=\mathbb{E}[\exp(tY)]=\mathbb{E}[\exp(tg(X))]=\mathbb{E}\biggl[\biggl(\frac{X}{1+X}\biggr)^t\biggr]=\int \biggl(\frac{x}{1+x}\biggr)^t f_X(x)\,dx$$

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