Transcription of Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 1 ...
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Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 1 ...
Chapter 5: JOINT PROBABILITY . DISTRIBUTIONS . Part 1: Sections to For both discrete and continuous random variables we will discuss the JOINT DISTRIBUTIONS (for two or more 's). Marginal DISTRIBUTIONS (computed from a JOINT distribution ). Conditional DISTRIBUTIONS ( P (Y = y|X = x)). Independence for 's X and Y. This is a good time to refresh your memory on double-integration. We will be using this skill in the upcom- ing lectures. 1. Recall a discrete PROBABILITY distribution (or pmf ) for a single X with the example be- x 0 1 2. f (x) Sometimes we're simultaneously interested in two or more variables in a random experiment. We're looking for a relationship between the two variables. Examples for discrete 's Year in college vs. Number of credits taken Number of cigarettes smoked per day vs. Day of the week Examples for continuous 's Time when bus driver picks you up vs. Quantity of caffeine in bus driver's system Dosage of a drug (ml) vs.
Joint Probability Density Function A joint probability density function for the continuous random variable X and Y, de-noted as fXY(x;y), satis es the following properties: 1. fXY(x;y) 0 for all x, y 2. R 1 1 R 1 1 fXY(x;y) dxdy= 1 3. For any region Rof 2-D space P((X;Y) 2R) = Z Z R fXY(x;y) dxdy For when the r.v.’s are continuous. 16
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