Joint and Marginal Probability in Random Variables
Explore the concepts of joint and marginal probability in random variables through examples like Buffon's Needle and Multivariate Gaussian distributions. Learn how to calculate probabilities and PDFs for different scenarios involving continuous random variables.
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Presentation Transcript
Tutorial 7: General Random Variables 3 Yitong Meng March 13, 2017 1
Revise - Joint PDF Two continuous random variables ?,? satisfying ? ?,? ? = ??,??,? ???? (?,?) ? for every subset ?
Revise - Marginal Probability ? ? ? = ? ? ?,? , ??,??,? ???? = ? The marginal PDF of ? is ??? = ??,??,? ??
Example: Buffons Needle A surface is ruled with parallel lines, which at distance ? from each other. Suppose we throw a needle of length ? randomly. What is the prob. that the needle will intersect one of the lines?
Example: Buffons Needle Assume ? < ? so that the needle cannot intersect two lines simultaneously. ?, the distance from the middle point of the needle and the nearest of the parallel lines ?, the acute angle formed by the needle and the lines
Example: Buffons Needle We model (?,?) with a uniform joint PDF so that 4 ??, 0, ?? ? 0,? 2 ??? ? [0,? ?? ?????? 2] ??,??,? =
Example: Buffons Needle The needle will intersect one of the lines if and only if ? ? 2sin?
Example: Buffons Needle So the probability of intersection is ? ? ? 2sin? = ??,??,? ???? ? ? 2sin ? ? 2sin ? ?/2 ?/2 4 4 ? 2 = ?? ???? = ?? sin??? 0 0 0 ? 2 0=2? =2? cos? ?? ??
Example: Multivariate Gaussian Let ? ??be a n-dimensional normal random variable. The PDF of ? is then 1 2? ?? 1? ? 1 ? 2 ??? = 1 2 2?
Example: Multivariate Gaussian The mean and variance ? ? = ? ? ? = (Recall) 1 2? ?? 1? ? 1 ? 2 ??? = 1 2 2?
Example: Multivariate Gaussian To draw: suppose we have ??~ (0,1) i.i.d and ? = ?1, ,?? We have ? + ??~ (?, ) where ???= .
Example: Multivariate Gaussian Element-wise variance ??= ??? ??,?? Degenerate case = ????(?1,?2, ,??)
Example: Multivariate Gaussian Fact: in degenerate case, elements of ? are independent Proof: As the coefficient of ???? is zero in ? ?? 1? ? , the PDF can be decomposed into the product of ?? part and ?? part.
Example: Multivariate Gaussian For conditional distribution: Gaussian variable condition on Gaussian variable is still Gaussian. We omit the proof.
