Expected Value (Joint)

Proposition

Let $X, Y$ be jointly distributed rvs with Joint Probability Mass Function (JPMF) $p (x, y)$ or Probability Distributions for Continuous Variables (PDF) $f (x, y)$ based on whether $X, Y$ are both discrete or continuous variables. Then the expected value of a function $h (X, Y)$ denoted by $E [h (X, Y)]$ or $μ_{h (X, Y)}$ is given by:

E [h (X, Y)] = {\begin{cases} \sum_{x} \sum_{y} h (x, y) \cdot p (x, y) & X, Y are discrete \\ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} h (x, y) \cdot f (x, y) d x d y & X, Y are continuous \end{cases}

Properties of Expected Value (Joint)

Linearity of Expectation

Let $X, Y$ be random variables. Then, for any function $h_{1}, h_{2}$ and any constants $a_{1}, a_{2}, b$ then:

E [a_{1} h_{1} (X, Y) + a_{2} h_{2} (X, Y) + b] = a_{1} E [h_{1} (X, Y)] + a_{2} E [h_{2} (X, Y)] + b

This comes directly from the linearity of expectation.

Independence implies splittable multiplication

Let $X, Y$ be independent random variables. If $h (X, Y) = g_{1} (X) \cdot g_{2} (Y)$ then:

E [h (X, Y)] = E [g_{1} (X) \cdot g_{2} (Y)] = E [g_{1} (X)] \cdot E [g_{2} (Y)]

Proof

For the continuous case and discrete case, there are pretty much carbon copies. Doing the continuous case:

\begin{aligned} E [h (X, Y)] & = E [g_{1} (X) \cdot g_{2} (Y)] \\ = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} g_{1} (x) \cdot g_{2} (y) \cdot f (x, y) d x d y \\ = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} g_{1} (x) \cdot g_{2} (y) \cdot f_{X} (x) \cdot f_{Y} (y) d x d y & X, Y independent \\ = (\int_{- \infty}^{\infty} g_{1} (x) f_{X} (x) d x) (\int_{- \infty}^{\infty} g_{2} (y) f_{Y} (y) d y) \\ = E [g_{1} (X)] E [g_{2} (Y)] \end{aligned}

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