Jointly Distributed Random Variables 1.4 2003/05/13 2008/03/20 23:01:52.256 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Mariyah Poonawala mariyah@rice.edu Kyle Evan Clarkson kclarks@gmail.com Kevin Duh kevinduh@rice.edu Elizabeth Gregory elizabeth.gregory@gmail.com Eileen Krause erkrause@rice.edu Jeffrey M Silverman JSilverman@astro.berkeley.edu 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 X x Y 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 β x α y p X Y α β or p X Y x y x y P X Y x y Since y 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 β p X Y x β and p Y y α p X Y α y Extending the ideas of conditional probabilities, the conditional probability density function p X | Y x | Y = y is defined (when p Y y 0 ) as p X | Y x | Y = y 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 the most important joint moment is the covariance: cov X Y X Y X Y where X Y 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 cov X Y σ X σ Y When two random variables are uncorrelated, their covariance and correlation coefficient equals zero so that X Y X Y . Statistically independent random variables are always uncorrelated, but uncorrelated random variables can be dependent. Let X be uniformly distributed over -1 1 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 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 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 conditional expected 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 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 l 1 L X l L X L X