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Transcript of 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables Figure 5-1 Joint...
![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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/1.jpg)
![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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/2.jpg)
![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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/3.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/4.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/5.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/6.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/7.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/8.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/9.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/10.jpg)
5-1 Two Discrete Random Variables
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5-1 Two Discrete Random Variables
5-1.3 Conditional Probability Distributions
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/13.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/14.jpg)
5-1 Two Discrete Random Variables5-1.4 Independence
Example 5-8
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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.
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5-1 Two Discrete Random Variables
5-1.4 Independence
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5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions
Definition
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5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions
Definition
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Example 5-11
5-2 Multiple Discrete Random Variables
Figure 5-5 Joint probability distribution of X1, X2, and X3.
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5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions
Mean and Variance from Joint Distribution
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5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions
Distribution of a Subset of Random Variables
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5-2 Multiple Discrete Random Variables5-2.1 Joint Probability Distributions
Conditional Probability Distributions
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5-2 Multiple Discrete Random Variables5-2.2 Multinomial Probability Distribution
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5-2 Multiple Discrete Random Variables
5-2.2 Multinomial Probability Distribution
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5-3 Two Continuous Random Variables
5-3.1 Joint Probability Distribution
Definition
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5-3 Two Continuous Random Variables
Figure 5-6 Joint probability density function for random variables X and Y.
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5-3 Two Continuous Random Variables
Example 5-15
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5-3 Two Continuous Random VariablesExample 5-15
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5-3 Two Continuous Random Variables
Figure 5-8 The joint probability density function of X and Y is nonzero over the shaded region.
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5-3 Two Continuous Random VariablesExample 5-15
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5-3 Two Continuous Random Variables
Figure 5-9 Region of integration for the probability that X < 1000 and Y < 2000 is darkly shaded.
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/33.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/34.jpg)
5-3 Two Continuous Random Variables
Example 5-16
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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.
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5-3 Two Continuous Random VariablesExample 5-16
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/38.jpg)
5-3 Two Continuous Random Variables
Example 5-16
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/40.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/41.jpg)
5-3 Two Continuous Random VariablesExample 5-17
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/43.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/44.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/45.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/46.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/47.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/48.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/49.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/50.jpg)
5-4 Multiple Continuous Random Variables
Distribution of a Subset of Random Variables
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/52.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/53.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/54.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/55.jpg)
5-5 Covariance and CorrelationExample 5-27
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/57.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/58.jpg)
5-5 Covariance and Correlation
Figure 5-13 Joint probability distributions and the sign of covariance between X and Y.
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5-5 Covariance and Correlation
Definition
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5-5 Covariance and Correlation
Example 5-29
Figure 5-14 Joint distribution for Example 5-29.
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5-5 Covariance and CorrelationExample 5-29 (continued)
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5-5 Covariance and Correlation
Example 5-31
Figure 5-16 Random variables with zero covariance from Example 5-31.
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5-5 Covariance and Correlation
Example 5-31 (continued)
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5-5 Covariance and CorrelationExample 5-31 (continued)
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5-5 Covariance and Correlation
Example 5-31 (continued)
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/67.jpg)
5-6 Bivariate Normal DistributionFigure 5-17. Examples of bivariate normal distributions.
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5-6 Bivariate Normal DistributionExample 5-33
Figure 5-18
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5-6 Bivariate Normal Distribution
Marginal Distributions of Bivariate Normal Random Variables
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5-6 Bivariate Normal Distribution
Figure 5-19 Marginal probability density functions of a bivariate normal distributions.
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/72.jpg)
5-6 Bivariate Normal Distribution
Example 5-34
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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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/74.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/75.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/76.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/77.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/78.jpg)
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.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649d985503460f94a83192/html5/thumbnails/79.jpg)