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I define a function as f[x_] := ...;which is in fact the PDF of a random variable $X$. The PDF if given by $f_X(x)$.

Let $Y$ is a random variable, which is the square of the random variable $X$, i.e., $Y=X^2$.

The PDF of $Y$ is expressed as $f_Y(y)$.

Let $I$ is a function of $Y$, which we express in MMA as I = ... (not as I[y_]:=)

I need to evaluate

$\exp\left(-\mathbb{E}[I]\right)$

How do I do it?

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    $\begingroup$ Please provide a complete minimal working example. "..." as a function body is useless to readers in general. $\endgroup$ – ciao May 3 '16 at 6:37
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I think you almost did right, but you need to give the distribution rather than the PDF of a distribution when evaluating the expectation.

For example:

f[x_] := Sqrt[2]/Pi/(1 + x^4)

distX = ProbabilityDistribution[f[x], {x, -Infinity, Infinity}];
distY = TransformedDistribution[X^2, X \[Distributed] distX];
distI = TransformedDistribution[2 Y + 1, Y \[Distributed] distY];

Exp[-Expectation[II, II \[Distributed] distI]]
(* 1/E^3 *)
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