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Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables, each with the probability density function $f(x)$, and let $\varphi(x_1,x_2,\ldots,x_n)$ be an integrable real-valued function of $n$ real variables.

What is the formula for the expectation $E[Y]$ of the random variable $Y=\varphi(X_1,X_2,\ldots,X_n)$?

I know that for a function of one variable, $g(x)$, there is the formula $$ E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx. $$ And what is the formula for $E[\varphi(X_1,X_2,\ldots,X_n)]$?

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    $\begingroup$ $$E[\varphi(X_1,X_2,\ldots,X_n)]=\int_{\mathbb R^n}\varphi(x_1,x_2,\ldots,x_n)f(x_1)f(x_2)\cdots f(x_n)dx_1dx_2\cdots dx_n$$ (Not in your book?) $\endgroup$ – Did Sep 28 '16 at 15:43

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