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Is their an easier way to find variance of function of random variable?

Till now what I am doing is first find probability density function of (function of random variable) then integrate over range.

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Assume $Y = g(X)$ be the function of RV $X$, then by using the following

$$E\{ g(X) \} = \int g(x) f_X(x) dx $$

variance of $Y$ can be computed without the computation of pdf $f_Y(y)$ as:

$$ \begin{align} \text{Var(Y)} &= E\{ (Y-\mu_Y)^2 \} = E\{ Y^2 \} - (\mu_Y)^2 \\ & = E\{ g^2(X) \} - E\{ g(X) \}^2 \\ & = \int g^2(x) f_x(x) dx - \left(\int g(x) f_X(x) dx \right)^2 \\ \end{align} $$

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