Chapter 5 Joint Probability Distributions and Random Samples  Jointly Distributed Random Variables.2 - Expected Values, Covariance, and Correlation.3 - Statistics and Their Distributions.4 - The Distribution of the Sample Mean.5 - The Distribution of a Linear Combination

discrete Suppose X and Y are two discrete random variables with pmfs f X (x) and f Y (y) respectively: discrete Suppose Z is a third discrete random variable that depends on X and Y, i.e., there exists a joint pmf for every ordered pair ( x, y)…

discrete Suppose X and Y are two discrete random variables with pmfs f X (x) and f Y (y) respectively:

discrete Suppose Z is a third discrete random variable that depends on X and Y, i.e., there exists a joint pmf for every ordered pair ( x, y)… discrete Suppose X and Y are two discrete random variables with pmfs f X (x) and f Y (y) respectively:

discrete Suppose Z is a third discrete random variable that depends on X and Y, i.e., there exists a joint pmf for every ordered pair ( x, y)… discrete Suppose X and Y are two discrete random variables with pmfs f X (x) and f Y (y) respectively:

discrete Suppose Z is a third discrete random variable that depends on X and Y, i.e., there exists a joint pmf for every ordered pair ( x, y)… discrete Suppose X and Y are two discrete random variables with pmfs f X (x) and f Y (y) respectively:

Def: X and Y are statistically independent if

i.e., each cell probability is equal to the product of its marginal probabilities.

Recall for X discrete… Probability Histogram

Recall for X discrete… continuous… As  x  0 and # rectangles  ∞, this “Riemann sum” approaches the area under the density curve, expressed as a definite integral. Probability Histogram

Joint Probability Mass Function

Probability Histogram Joint Probability Mass Function

Similarly…

Joint Probability Mass Function Joint Probability Density Function Volume under density f(x, y) over A. “area element” Area A

Joint Probability Density Function Example: Uniform Distribution

Joint Probability Density Function A Example: Uniform Distribution

Joint Probability Density Function A Example:

Joint Probability Density Function A Example: A

A Joint Probability Density Function Example:

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Joint Probability Mass Function

Joint Probability Density Function

Joint Probability Mass Function Joint Probability Density Function

Joint Probability Mass Function Joint Probability Density Function

A Example (revisted):

Joint Probability Density Function A Example (revisted):

Joint Probability Density Function Example (revisted): A

Joint Probability Mass Function Joint Probability Density Function

A Exercise:

Joint Probability Density Function Exercise: A

Joint Probability Density Function Exercise: A

Joint Probability Density Function X and Y are not independent if A is not a standard rectangle! Exercise:

Volume under density f(x, y) over A. Joint Probability Density Function Area A “Hypervolume” under density f over A. Definition of statistical independence of X and Y can be extended to any number of variables.