Transcription of Chapter 4 Multivariate distributions
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Chapter 4 Multivariate distributions
RS 4 Multivariate Distributions1 Chapter 4 Multivariate distributionsk 2 Multivariate DistributionsAll the results derived for the bivariate case can be generalized to n RV. The joint CDF of X1, X2, .., Xk will have the form: P(x1, x2, .., xk) when the RVs are discreteF(x1, x2, .., xk) when the RVs are continuousRS 4 Multivariate Distributions2 Joint Probability FunctionDefinition: Joint Probability FunctionLet X1, X2, .., Xk denote k discrete random variables, then p(x1, x2, .., xk) is joint probability function of X1, X2, .., Xk if 112. ,,1nnxxpxx 11. 0,,1npxx 113. ,,,,nnPXXApxx 1,,nxxA Definition: Joint density function Let X1, X2, .., Xk denote k continuous random variables, then f(x1, x2, .., xk) = n/ x1, x2, .., xkF(x1, x2, .., xk)is the joint density function of X1, X2.
RS – 4 – Multivariate Distributions 3 Example: The Multinomial distribution Suppose that we observe an experiment that has k possible outcomes {O1, O2, …, Ok} independently n times.Let p1, p2, …, pk denote probabilities of O1, O2, …, Ok respectively. Let Xi denote the number of times that outcome Oi occurs in the n repetitions of the experiment.
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