Tutorial 5:General Random Variables 1 MENG Yitong ytmeng@cse.cuhk.edu.hk 27,February,2016 1
Tutorial 5: General Random Variables 1 MENG Yitong ytmeng@cse.cuhk.edu.hk 27, February, 2016 1
Outline Continuous random variables,PDFs,CDFs Uniform distributions Exponential distributions Normal distributions Laplace distributions 2
Outline • Continuous random variables, PDFs, CDFs • Uniform distributions • Exponential distributions • Normal distributions • Laplace distributions 2
Continuous r.v.and PDFs A random variable X is called continuous if there is a functionfx>0, called the probability density function of X,or PDF,s.t. P(x eB)f(dx for every subset B R. In particular,when B [a,b], P(a≤X≤b)=fx(x)dx is the area under the graph of PDF. 3
Continuous r.v. and PDFs • A random variable 𝑋 is called continuous if there is a function 𝑓𝑋 ≥ 0, called the probability density function of 𝑋, or PDF, s.t. 𝑃(𝑋 ∈ 𝐵) = 𝑓𝑋 𝑥 𝑑𝑥 𝐵 for every subset 𝐵 ⊆ ℝ. • In particular, when 𝐵 = [𝑎, 𝑏], 𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) = 𝑓𝑋 𝑥 𝑑𝑥 𝑏 𝑎 is the area under the graph of PDF. 3
Continuous r.v.and PDFs PDF fx(z) Sample space b 2 Event fas Xsb) 4
Continuous r.v. and PDFs 4
Cumulative Distribution Function The cumulative distribution function,or CDF,of a random variable X is Fx(x)=P(X≤x) 了∑ksx Px(x),( discrete fx(x)dx,continuous ·The CDF Fx(x)“accumulates"probability“upto"the valuex
Cumulative Distribution Function • The cumulative distribution function, or CDF, of a random variable 𝑋 is 𝐹𝑋 𝑥 = 𝑃 𝑋 ≤ 𝑥 = 𝑘≤𝑥 𝑝𝑋 𝑥 , discrete 𝑓𝑋 𝑥 𝑑𝑥 𝑥 −∞ , continuous • The CDF 𝐹𝑋 𝑥 “accumulates” probability “up to” the value 𝑥