Download presentation
Presentation is loading. Please wait.
1
Probability theory 2010 Conditional distributions Conditional probability: Conditional probability mass function: Discrete case Conditional probability mass function: Continuous case
2
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
3
Probability theory 2010 Using conditional probability mass functions to compute joint and marginal densities Discrete case Continuous case
4
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
5
Probability theory 2010 Conditional expectation Discrete case Continuous case Notation
6
Probability theory 2010 Conditional expectation - rules
7
Probability theory 2010 Calculation of expected values through conditioning Discrete case Continuous case General formula
8
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
9
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
10
Probability theory 2010 Variance decomposition in linear regression
11
Probability theory 2010 Proof of the variance decomposition We shall prove that It can easily be seen that
12
Probability theory 2010 Regression and prediction Regression function: Theorem: The regression function is the best predictor of Y based on X Proof:
13
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
14
Probability theory 2010 Expected quadratic prediction error of the best linear predictor Theorem: Proof: ……. Ordinary linear regression
15
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
16
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.