Expectation

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Definition

Definition

Let $X$ be a positive real random variable, then its expectation is defined by the quantity, may be infinite,

$$\mathbb{E}[X] := \int_\Omega X(\omega)d\mathbb{P}(\omega) = \int_\mathbb{R}xd\mathbb{P}_X(x).$$

Results

The following proposition calculates the expectation of a random variable without knowing its density.

Proposition : Transfert function

Let $X$ be a random variable with density $f$, and $h$ a real measurable function. Assume that

$$\int_\mathbb{R}|h(x)|f(x)dx < +\infty$$

Then, if we set Y = h(X), Y is an integrable random variable and

$$\mathbb{E}[Y] = \mathbb{E}[h(X)] = \int_\mathbb{R}h(x)f(x)dx$$