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5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

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Page 1: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1
Page 2: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1
Page 3: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

Example 5-1

Page 4: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

Page 5: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

5-1.1 Joint Probability Distributions

Page 6: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

5-1.2 Marginal Probability Distributions

• The individual probability distribution of a random variable is referred to as its marginal probability distribution.

• In general, the marginal probability distribution of X can be determined from the joint probability distribution of X and other random variables. For example, to determine P(X = x), we sum P(X = x, Y = y) over all points in the range of (X, Y ) for which X = x. Subscripts on the probability mass functions distinguish between the random variables.

Page 7: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

Example 5-2

Page 8: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

Figure 5-2 Marginal probability distributions of X and Y from Figure 5-1.

Page 9: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

Definition

Page 10: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

Page 11: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

5-1.3 Conditional Probability Distributions

Page 12: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

5-1.3 Conditional Probability Distributions

Page 13: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

Example 5-6

Figure 5-3 Conditional probability distributions of Y given X, fY|x(y) in Example 5-6.

Page 14: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables5-1.4 Independence

Example 5-8

Page 15: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

Example 5-8

Figure 5-4 (a)Joint and marginal probability distributions of X and Y in Example 5-8. (b) Conditional probability distribution of Y given X = x in Example 5-8.

Page 16: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-1 Two Discrete Random Variables

5-1.4 Independence

Page 17: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions

Definition

Page 18: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions

Definition

Page 19: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

Example 5-11

5-2 Multiple Discrete Random Variables

Figure 5-5 Joint probability distribution of X1, X2, and X3.

Page 20: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions

Mean and Variance from Joint Distribution

Page 21: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions

Distribution of a Subset of Random Variables

Page 22: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions

Conditional Probability Distributions

Page 23: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables5-2.2 Multinomial Probability Distribution

Page 24: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-2 Multiple Discrete Random Variables

5-2.2 Multinomial Probability Distribution

Page 25: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

5-3.1 Joint Probability Distribution

Definition

Page 26: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Figure 5-6 Joint probability density function for random variables X and Y.

Page 27: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-15

Page 28: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random VariablesExample 5-15

Page 29: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Figure 5-8 The joint probability density function of X and Y is nonzero over the shaded region.

Page 30: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random VariablesExample 5-15

Page 31: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Figure 5-9 Region of integration for the probability that X < 1000 and Y < 2000 is darkly shaded.

Page 32: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

5-3.2 Marginal Probability Distributions

Definition

Page 33: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random VariablesMean and Variance from Joint Distribution

Page 34: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-16

Page 35: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Figure 5-10 Region of integration for the probability that Y < 2000 is darkly shaded and it is partitioned into two regions with x < 2000 and and x > 2000.

Page 36: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random VariablesExample 5-16

Page 37: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-16

Page 38: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-16

Page 39: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

5-3.3 Conditional Probability Distributions

Definition

Page 40: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

5-3.3 Conditional Probability Distributions

Page 41: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random VariablesExample 5-17

Page 42: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-17

Figure 5-11 The conditional probability density function for Y, given that x = 1500, is nonzero over the solid line.

Page 43: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Definition

Page 44: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

5-3.4 Independence

Definition

Page 45: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-19

Page 46: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-3 Two Continuous Random Variables

Example 5-21

Page 47: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Example 5-23

Page 48: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Definition

Page 49: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Mean and Variance from Joint Distribution

Page 50: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Distribution of a Subset of Random Variables

Page 51: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Conditional Probability Distribution

Definition

Page 52: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Example 5-26

Page 53: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-4 Multiple Continuous Random Variables

Example 5-26

Page 54: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Definition

Page 55: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and CorrelationExample 5-27

Page 56: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and CorrelationExample 5-27

Figure 5-12 Joint distribution of X and Y for Example 5-27.

Page 57: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Definition

Page 58: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Figure 5-13 Joint probability distributions and the sign of covariance between X and Y.

Page 59: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Definition

Page 60: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Example 5-29

Figure 5-14 Joint distribution for Example 5-29.

Page 61: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and CorrelationExample 5-29 (continued)

Page 62: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Example 5-31

Figure 5-16 Random variables with zero covariance from Example 5-31.

Page 63: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Example 5-31 (continued)

Page 64: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and CorrelationExample 5-31 (continued)

Page 65: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-5 Covariance and Correlation

Example 5-31 (continued)

Page 66: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal DistributionDefinition

Page 67: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal DistributionFigure 5-17. Examples of bivariate normal distributions.

Page 68: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal DistributionExample 5-33

Figure 5-18

Page 69: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal Distribution

Marginal Distributions of Bivariate Normal Random Variables

Page 70: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal Distribution

Figure 5-19 Marginal probability density functions of a bivariate normal distributions.

Page 71: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal Distribution

Page 72: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-6 Bivariate Normal Distribution

Example 5-34

Page 73: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-7 Linear Combinations of Random Variables

Definition

Mean of a Linear Combination

Page 74: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-7 Linear Combinations of Random Variables

Variance of a Linear Combination

Page 75: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-7 Linear Combinations of Random Variables

Example 5-36

Page 76: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-7 Linear Combinations of Random Variables

Mean and Variance of an Average

Page 77: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-7 Linear Combinations of Random Variables

Reproductive Property of the Normal Distribution

Page 78: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

5-7 Linear Combinations of Random Variables

Example 5-37

Page 79: 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1

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