# 1.2 Jointly distributed random variables

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Two (or more) random variables can be defined over the same sample space. Just as with jointly defined events, the joint distribution function is easily defined.

$P(X, , Y, x, y)\equiv (\{X\le x\}\cap \{Y\le y\})$
The joint probability density function $p(X, , Y, x, y)$ is related to the distribution function via double integration.
$P(X, , Y, x, y)=\int_{()} \,d$ x y p X Y
or $p(X, , Y, x, y)=\frac{\partial^{2}P(X, , Y, x, y)}{\partial x\partial y}$ Since $\lim_{y\to }y\to$ P X Y x y P X x , the so-called marginal density functions can be related to the joint density function.
$p(X, x)=\int_{()} \,d$ p X Y x
and $p(Y, y)=\int_{()} \,d$ p X Y y

Extending the ideas of conditional probabilities, the conditional probability density function ${p}_{X|Y}(x|Y=y)$ is defined (when $p(Y, y)\neq 0$ ) as

${p}_{X|Y}(x|Y=y)=\frac{p(X, , Y, x, y)}{p(Y, y)}$
Two random variables are statistically independent when ${p}_{X|Y}(x|Y=y)=p(X, x)$ , which is equivalent to the condition that the joint density function is separable: $p(X, , Y, x, y)=p(X, x)p(Y, y)$ .

For jointly defined random variables, expected values are defined similarly as with single random variables. Probably themost important joint moment is the covariance :

$\mathrm{cov}(X, Y)\equiv (XY)-(X)(Y)$
where $(XY)=\int_{()} \,d y$ x x y p X Y x y Related to the covariance is the (confusingly named) correlation coefficient : the covariance normalized by the standard deviations of the component random variables. ${p}_{X,Y}=\frac{\mathrm{cov}(X, Y)}{{}_{X}{}_{Y}}$ When two random variables are uncorrelated , their covariance and correlation coefficient equals zero so that $(XY)=(X)(Y)$ . Statistically independent random variables are always uncorrelated, but uncorrelated random variables can bedependent.
Let $X$ be uniformly distributed over $\left[-1 , 1\right]$ and let $Y=X^{2}$ . The two random variables are uncorrelated, but are clearly not independent.

A conditional expected value is the mean of the conditional density.

$(Y, X)=\int_{()} \,d x$ p X | Y x | Y = y
Note that the conditional expected value is now a function of $Y$ and is therefore a random variable. Consequently, it too has an expected value, which is easily evaluated to be the expected value of $X$ .
$((Y, X))=\int_{()} \,d y$ x x p X | Y x | Y = y p Y y X
More generally, the expected value of a function of two random variables can be shown to be the expected value of a conditionalexpected value: $(f(X, Y))=((Y, f(X, Y)))$ . This kind of calculation is frequently simpler to evaluate than trying to find the expected value of $f(X, Y)$ "all at once." A particularly interesting example of this simplicity is the random sum of random variables . Let $L$ be a random variable and $\{{X}_{l}\}$ a sequence of random variables. We will find occasion to consider the quantity $\sum_{l=1}^{L} {X}_{l}$ . Assuming that each component of the sequence has the same expected value $(X)$ , the expected value of the sum is found to be
$({S}_{L})=((L, \sum_{l=1}^{L} {X}_{l}))=(L(X))=(L)(X)$

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