logging in or signing up Joint Probability Distribution aSGuest2427 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 5520 Category: Education License: All Rights Reserved Like it (3) Dislike it (0) Added: November 03, 2008 This Presentation is Public Favorites: 1 Presentation Description No description available. Comments Posting comment... By: aakhdr1 (31 month(s) ago) Dear Presenter; I thanks you for this effort, so please allow me to download it .Thanks. Saving..... Post Reply Close Saving..... Edit Comment Close By: aakhdr1 (31 month(s) ago) good this file Saving..... Post Reply Close Saving..... Edit Comment Close By: yasmin_az (36 month(s) ago) can i have a copy of the presentation slides. tq Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables : 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 Two Discrete Random Variables 5-1.1 Joint Probability Distributions 5-1 Two Discrete Random Variables : 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. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Example 5-2 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Figure 5-2 Marginal probability distributions of X and Y from Figure 5-1. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Definition 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.3 Conditional Probability Distributions 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.3 Conditional Probability Distributions Slide 13: Example 5-6 Figure 5-3 Conditional probability distributions of Y given X, fY|x(y) in Example 5-6. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.4 Independence Example 5-8 Slide 15: 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. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.4 Independence 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Definition 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Definition 5-2 Multiple Discrete Random Variables : Example 5-11 5-2 Multiple Discrete Random Variables Figure 5-5 Joint probability distribution of X1, X2, and X3. 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Mean and Variance from Joint Distribution 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Distribution of a Subset of Random Variables 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Conditional Probability Distributions 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.2 Multinomial Probability Distribution 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.2 Multinomial Probability Distribution 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.1 Joint Probability Distribution Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Figure 5-6 Joint probability density function for random variables X and Y. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-15 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-15 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Figure 5-8 The joint probability density function of X and Y is nonzero over the shaded region. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-15 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Figure 5-9 Region of integration for the probability that X < 1000 and Y < 2000 is darkly shaded. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.2 Marginal Probability Distributions Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Mean and Variance from Joint Distribution 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 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. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.3 Conditional Probability Distributions Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.3 Conditional Probability Distributions 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-17 5-3 Two Continuous Random Variables : 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. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.4 Independence Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-19 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-21 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Example 5-23 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Definition 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Mean and Variance from Joint Distribution 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Distribution of a Subset of Random Variables 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Conditional Probability Distribution Definition 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Example 5-26 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Example 5-26 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Definition 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-27 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-27 Figure 5-12 Joint distribution of X and Y for Example 5-27. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Definition 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Figure 5-13 Joint probability distributions and the sign of covariance between X and Y. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Definition 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-29 Figure 5-14 Joint distribution for Example 5-29. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-29 (continued) 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 Figure 5-16 Random variables with zero covariance from Example 5-31. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 (continued) 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 (continued) 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 (continued) 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Definition 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Figure 5-17. Examples of bivariate normal distributions. 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Example 5-33 Figure 5-18 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Marginal Distributions of Bivariate Normal Random Variables 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Figure 5-19 Marginal probability density functions of a bivariate normal distributions. 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Example 5-34 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Definition Mean of a Linear Combination 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Variance of a Linear Combination 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Example 5-36 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Mean and Variance of an Average 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Reproductive Property of the Normal Distribution 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Example 5-37 You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
Joint Probability Distribution aSGuest2427 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 5520 Category: Education License: All Rights Reserved Like it (3) Dislike it (0) Added: November 03, 2008 This Presentation is Public Favorites: 1 Presentation Description No description available. Comments Posting comment... By: aakhdr1 (31 month(s) ago) Dear Presenter; I thanks you for this effort, so please allow me to download it .Thanks. Saving..... Post Reply Close Saving..... Edit Comment Close By: aakhdr1 (31 month(s) ago) good this file Saving..... Post Reply Close Saving..... Edit Comment Close By: yasmin_az (36 month(s) ago) can i have a copy of the presentation slides. tq Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Example 5-1 5-1 Two Discrete Random Variables : 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 Two Discrete Random Variables 5-1.1 Joint Probability Distributions 5-1 Two Discrete Random Variables : 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. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Example 5-2 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Figure 5-2 Marginal probability distributions of X and Y from Figure 5-1. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables Definition 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.3 Conditional Probability Distributions 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.3 Conditional Probability Distributions Slide 13: Example 5-6 Figure 5-3 Conditional probability distributions of Y given X, fY|x(y) in Example 5-6. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.4 Independence Example 5-8 Slide 15: 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. 5-1 Two Discrete Random Variables : 5-1 Two Discrete Random Variables 5-1.4 Independence 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Definition 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Definition 5-2 Multiple Discrete Random Variables : Example 5-11 5-2 Multiple Discrete Random Variables Figure 5-5 Joint probability distribution of X1, X2, and X3. 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Mean and Variance from Joint Distribution 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Distribution of a Subset of Random Variables 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.1 Joint Probability Distributions Conditional Probability Distributions 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.2 Multinomial Probability Distribution 5-2 Multiple Discrete Random Variables : 5-2 Multiple Discrete Random Variables 5-2.2 Multinomial Probability Distribution 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.1 Joint Probability Distribution Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Figure 5-6 Joint probability density function for random variables X and Y. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-15 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-15 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Figure 5-8 The joint probability density function of X and Y is nonzero over the shaded region. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-15 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Figure 5-9 Region of integration for the probability that X < 1000 and Y < 2000 is darkly shaded. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.2 Marginal Probability Distributions Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Mean and Variance from Joint Distribution 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 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. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-16 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.3 Conditional Probability Distributions Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.3 Conditional Probability Distributions 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-17 5-3 Two Continuous Random Variables : 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. 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables 5-3.4 Independence Definition 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-19 5-3 Two Continuous Random Variables : 5-3 Two Continuous Random Variables Example 5-21 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Example 5-23 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Definition 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Mean and Variance from Joint Distribution 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Distribution of a Subset of Random Variables 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Conditional Probability Distribution Definition 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Example 5-26 5-4 Multiple Continuous Random Variables : 5-4 Multiple Continuous Random Variables Example 5-26 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Definition 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-27 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-27 Figure 5-12 Joint distribution of X and Y for Example 5-27. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Definition 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Figure 5-13 Joint probability distributions and the sign of covariance between X and Y. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Definition 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-29 Figure 5-14 Joint distribution for Example 5-29. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-29 (continued) 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 Figure 5-16 Random variables with zero covariance from Example 5-31. 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 (continued) 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 (continued) 5-5 Covariance and Correlation : 5-5 Covariance and Correlation Example 5-31 (continued) 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Definition 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Figure 5-17. Examples of bivariate normal distributions. 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Example 5-33 Figure 5-18 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Marginal Distributions of Bivariate Normal Random Variables 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Figure 5-19 Marginal probability density functions of a bivariate normal distributions. 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution 5-6 Bivariate Normal Distribution : 5-6 Bivariate Normal Distribution Example 5-34 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Definition Mean of a Linear Combination 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Variance of a Linear Combination 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Example 5-36 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Mean and Variance of an Average 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Reproductive Property of the Normal Distribution 5-7 Linear Combinations of Random Variables : 5-7 Linear Combinations of Random Variables Example 5-37