Example:maximum of several random variables Take a test three times with score in {1,..,10} The final score is the maximum of the scores X max(X1,X2,X3) Each X;takes values {1,..,10}eually likely, and different Xi's are independent. The CDF of the final score X is ■Fx(k)=P(X≤k) =P(X1≤k)P(X2≤k)P(X3≤k) =(k/10)3
Example: maximum of several random variables Take a test three times with score in {1, . . , 10} The final score is the maximum of the scores 𝑋 = max(𝑋1, 𝑋2, 𝑋3) Each 𝑋𝑖 takes values {1, . . , 10} eually likely, and different 𝑋𝑖 ’s are independent. The CDF of the final score 𝑋 is 𝐹𝑋 𝑘 = 𝑃 𝑋 ≤ 𝑘 = 𝑃 𝑋1 ≤ 𝑘 𝑃 𝑋2 ≤ 𝑘 𝑃 𝑋3 ≤ 𝑘 = (𝑘/10) 3
Example:maximum of several random variables Take a test three times with score in {1,..,10} The final score is the maximum of the scores X=max(X1,X2,X3) Each X;takes values {1,..,10}eually likely, and different Xi's are independent. The PDF of the final score X is Px(k)=Fx(k)-Fx(k1)=())
Example: maximum of several random variables Take a test three times with score in {1, . . , 10} The final score is the maximum of the scores 𝑋 = max(𝑋1, 𝑋2, 𝑋3) Each 𝑋𝑖 takes values {1, . . , 10} eually likely, and different 𝑋𝑖 ’s are independent. The PDF of the final score 𝑋 is 𝑃𝑋 𝑘 = 𝐹𝑋 𝑘 − 𝐹𝑋 𝑘 − 1 = 𝑘 10 3 − 𝑘−1 10 3
Example:Geometric and Exponential CDN for geometric random variable: e)）-2=1p1-p=p片 =1-(1-p)for n=1,2, CDN for exponential random variable: When x≤0：Fexp(x)=P(X≤0)=0 ·Nhen x≥0：Fexp(x)=J0e-tdt =-el6=1-ex
Example: Geometric and Exponential CDN for geometric random variable: 𝐹𝑔𝑒𝑜 𝑛 = 𝑝(1 − 𝑝) 𝑛 𝑘 𝑘=1 = 𝑝 1−(1−𝑝) 𝑛 1−(1−𝑝) = 1 − (1 − 𝑝) 𝑛 for 𝑛 = 1,2, … CDN for exponential random variable: When 𝑥 ≤ 0: 𝐹𝑒𝑥𝑝 𝑥 = 𝑃 𝑋 ≤ 0 = 0 When 𝑥 ≥ 0: 𝐹𝑒𝑥𝑝 𝑥 = 𝜆𝑒 −𝜆𝑡𝑑𝑡 𝑥 0 = −𝑒 −𝜆𝑡 𝑥 0 = 1 − 𝑒 −𝜆𝑥
Example:Geometric and Exponential Fgeo(n)=1-(1-p)",Fexp(x)=1-e-Ax. If e-16 =1-p,then Fexp(n8)=Fgeo(n). Exponential CDF 1-e-Ax 1 Geometric CDF:1-(1-p)"with p=1-e-48
Example: Geometric and Exponential 𝐹𝑔𝑒𝑜 𝑛 = 1 − (1 − 𝑝) 𝑛 , 𝐹𝑒𝑥𝑝 𝑥 = 1 − 𝑒 −𝜆𝑥 . If 𝑒 −𝜆𝛿 = 1 − 𝑝, then 𝐹𝑒𝑥𝑝 𝑛𝛿 = 𝐹𝑔𝑒𝑜 𝑛
Content Continuous Random Variables and PDFs Cumulative Distribution Functions Normal Random Variables Joint PDFs of Multiple Random Variables Conditioning The Continuous Bayes'Rule
Content Continuous Random Variables and PDFs Cumulative Distribution Functions Normal Random Variables Joint PDFs of Multiple Random Variables Conditioning The Continuous Bayes’ Rule