香港中文大学：《Probability and Statistics for Engineers》课程教学资源（辅导材料）Tutorial 7：General Random Variables 3

Tutorial 7:General Random Variables 3 Yitong Meng March 13,2017 1

Tutorial 7: General Random Variables 3 Yitong Meng March 13, 2017 1

Revise-Joint PDF Two continuous random variables X,Y satisfying P(xY)∈B)=∬kv(x,y)dxdy X,y)∈B for every subset B

Revise - Joint PDF  Two continuous random variables 𝑋, 𝑌 satisfying 𝑃 𝑋, 𝑌 ∈ 𝐵 = ඵ (𝑥,𝑦)∈𝐵 𝑓𝑋,𝑌 𝑥, 𝑦 𝑑𝑥𝑑𝑦 for every subset 𝐵

Revise-Marginal Probability ·P(X∈A)=P(X∈A,Y∈(-∞，∞) =∫Ck0r 00 The marginal PDF of X is 00 fx(x)=fx.r(x,y)dy

Revise - Marginal Probability  𝑃 𝑋 ∈ 𝐴 = 𝑃 𝑋 ∈ 𝐴, 𝑌 ∈ −∞, ∞ = න 𝐴 න −∞ ∞ 𝑓𝑋,𝑌 𝑥, 𝑦 𝑑𝑦 𝑑𝑥  The marginal PDF of 𝑋 is 𝑓𝑋 𝑥 = න −∞ ∞ 𝑓𝑋,𝑌 𝑥, 𝑦 𝑑𝑦

Example:Buffon's Needle A surface is ruled with parallel lines,which at distance d from each other. Suppose we throw a needle of length l randomly. d What is the prob.that the needle will intersect one of the lines?

Example: Buffon’s Needle  A surface is ruled with parallel lines, which at distance 𝑑 from each other.  Suppose we throw a needle of length 𝑙 randomly.  What is the prob. that the needle will intersect one of the lines?

Example:Buffon's Needle Assume l<d so that the needle cannot intersect two lines simultaneously. X,the distance from the 2 middle point of the needle and the nearest of the parallel lines the acute angle formed by the needle and the lines

Example: Buffon’s Needle  Assume 𝑙 < 𝑑 so that the needle cannot intersect two lines simultaneously.  𝑋, the distance from the middle point of the needle and the nearest of the parallel lines  𝜗, the acute angle formed by the needle and the lines

Example:Buffon's Needle We model (with a uniform joint PDF so that 4 x(x,8)=πd' xcmdc旺 （0, otherwise

Example: Buffon’s Needle  We model (𝑋, 𝜗) with a uniform joint PDF so that 𝑓𝑋,𝜗 𝑥, 𝜃 = ൞ 4 𝜋𝑑 , 𝑖𝑓 𝑥 ∈ 0, 𝑑 2 𝑎𝑛𝑑 𝜃 ∈ [0, 𝜋 2 ] 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Example:Buffon's Needle d The needle will intersect one of the lines if and only if X≤ sin

Example: Buffon’s Needle  The needle will intersect one of the lines if and only if 𝑋 ≤ 𝑙 2 sin 𝜗

Example:Buffon's Needle So the probability of intersection is psm-瓜 fx.(x,)dxde sin 0 4 dxd0 (-6s月 21

Example: Buffon’s Needle  So the probability of intersection is 𝑃 𝑋 ≤ 𝑙 2 sin 𝜗 = ඵ 𝑋≤ 𝑙 2 sin 𝜃 𝑓𝑋,𝜗 𝑥, 𝜃 𝑑𝑥𝑑𝜃 = 4 𝜋𝑑 න 0 𝜋/2 න 0 𝑙 2 sin 𝜃 𝑑𝑥𝑑𝜃 = 4 𝜋𝑑 න 0 𝜋/2 𝑙 2 sin 𝜃 𝑑𝜃 = 2𝑙 𝜋𝑑 − cos 𝜃 อ 𝜋 2 0 = 2𝑙 𝜋𝑑

Example:Multivariate Gaussian ■LetX∈RT be a n-dimensional normal random variable.The PDF of X is then 1 -2(x-0Tz-1(x-)） fx(x)= n 1 (2π)22

Example: Multivariate Gaussian  Let 𝑋 ∈ 𝑅 𝑛 be a n-dimensional normal random variable. The PDF of 𝑋 is then 𝑓𝑋 𝑥 = 1 2𝜋 𝑛 2 Σ 1 2 − 1 2 𝑥−𝜇 𝑇Σ −1 𝑥−𝜇

Example:Multivariate Gaussian -The mean and variance E(X) =u V(X)=Σ (Recall) 1 -2x-w)TE-1(x-w) fx(x)= (2π)z|2

Example: Multivariate Gaussian  The mean and variance 𝐸 𝑋 = 𝜇 𝑉 𝑋 = Σ (Recall) 𝑓𝑋 𝑥 = 1 2𝜋 𝑛 2 Σ 1 2 − 1 2 𝑥−𝜇 𝑇Σ −1 𝑥−𝜇