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12 Joint probability distributions

# 12 Joint probability distributions - University of...

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University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Joint probability distributions So far we have considered only distributions with one random variable. There are many problems that involve two or more random variables. We need to examine how these ran- dom variables are distributed together (“jointly”). There are discrete and continuous joint distributions. Discrete: Here is an Example: Let X be the number of puppies born, and Y be the number of puppies survived for a certain breed of dog. Suppose the following table describes the distribution of X and Y : X 3 4 5 0 0.31 0.21 0.21 0.73 Y 1 0.03 0.04 0.05 0.12 2 0.02 0.03 0.04 0.09 3 0.01 0.02 0.03 0.06 0.37 0.30 0.33 1.0 Notation: Joint (or bivariate) probability distribution of X and Y : f XY ( x, y ) = P ( X = x, Y = y ) Example: Always X x X y f XY ( x, y ) = 1 1

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Marginal distribution of X : f X ( x ) = X y f XY ( x, y ) Example: Marginal distribution of Y : f Y ( y ) = X x f XY ( x, y ) Example: Conditional probability function: f Y | X ( y | x ) = P ( Y = y | X = x ) = P ( Y = y, X = x ) P ( X = x ) Or f Y | X ( y | x ) = f XY ( x, y ) f X ( x ) Example: Joint cumulative distribution function: F XY ( x, y ) = P ( X x, Y y ) Example:
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12 Joint probability distributions - University of...

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