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Probability theory 2010 Conditional distributions Conditional probability: Conditional probability mass function: Discrete case Conditional probability mass function: Continuous case

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Probability theory 2010 Conditional probability mass functions - examples Throwing two dice Let Z 1 = the number on the first die Let Z 2 = the number on the second die Set Y = Z 1 and X = Z 1 +Z 2 Radioactive decay Let X = the number of atoms decaying within 1 unit of time Let Y = the time of the first decay

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Probability theory 2010 Using conditional probability mass functions to compute joint and marginal densities Discrete case Continuous case

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Probability theory 2010 Using conditional probability mass functions to compute marginal densities - Gibb’s sampler Suppose that for two random variables X and Y we know Then Moreover, the solution to this fixed-point equation can be obtained by successively sampling

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Probability theory 2010 Conditional expectation Discrete case Continuous case Notation

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Probability theory 2010 Conditional expectation - rules

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Probability theory 2010 Calculation of expected values through conditioning Discrete case Continuous case General formula

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Probability theory 2010 Calculation of expected values through conditioning - example Primary and secondary events Let N denote the number of primary events Let X 1, X 2, … denote the number of secondary events for each primary event Set Y = X 1 + X 2 + … + X N Assume that X 1, X 2, … are i.i.d. and independent of N

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Probability theory 2010 Calculation of variances through conditioning Variation in the expected value of Y induced by variation in X Average remaining variation in Y after X has been fixed

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Probability theory 2010 Variance decomposition in linear regression

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Probability theory 2010 Proof of the variance decomposition We shall prove that It can easily be seen that

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Probability theory 2010 Regression and prediction Regression function: Theorem: The regression function is the best predictor of Y based on X Proof:

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Probability theory 2010 Best linear predictor Theorem: The best linear predictor of Y based on X is Proof: Differentiate with respect to the parameters of the linear predictor. Ordinary linear regression

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Probability theory 2010 Expected quadratic prediction error of the best linear predictor Theorem: Proof: ……. Ordinary linear regression

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Probability theory 2010 Martingales The sequence X 1, X 2,… is called a martingale if Example 1: Partial sums of independent variables with mean zero Example 2: Gambler’s fortune if he doubles the stake as long as he loses and leaves as soon as he wins

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Probability theory 2010 Exercises: Chapter II 2.8, 2.11, 2.23, 2.35, 2.37 Use conditional distributions/probabilities to explain why the envelop-rejection method works

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