Tutorial 4:Discrete Random Variables 2 Baoxiang Wang bxwang@cse Spring 2017
Tutorial 4: Discrete Random Variables 2 Baoxiang Wang bxwang@cse Spring 2017 1
Geometric Random variables The PMF of a geometric random variable is px(k)=(1-p)k-1p ·EXX>1]=1+E[X]
Geometric Random Variables • The PMF of a geometric random variable is 𝑝𝑋 𝑘 = (1 − 𝑝) 𝑘−1𝑝 • 𝐸 𝑋 𝑋 > 1 = 1 + 𝐸[𝑋] 2
Geometric Random variables 。Fork>1, P(X=kx >1)= P(X=k 2=_(1-p)k-1p (1-p)k-1p P(X>1) 280-2(1-p)k-1p -p(1-p)1-4-p 1 =(1-p)k-2p=px(k-1) ·EXx>1]=∑K=2kg(X=kX>1) -∑px-1)=∑k+1pr侧=∑px+∑x个 =子E[X] k=1 k=1 3
Geometric Random Variables • For 𝑘 > 1, 𝑃 𝑋 = 𝑘 𝑋 > 1 = 𝑃(𝑋=𝑘) 𝑃(𝑋>1) = (1−𝑝) 𝑘−1𝑝 σ𝑘=2 ∞ (1−𝑝) 𝑘−1𝑝 = (1−𝑝) 𝑘−1𝑝 𝑝(1−𝑝)∙ 1 1−(1−𝑝) = (1 − 𝑝) 𝑘−2𝑝 = 𝑝𝑋(𝑘 − 1) • 𝐸 𝑋 𝑋 > 1 = σ𝑘=2 ∞ 𝑘𝑃 𝑋 = 𝑘 𝑋 > 1 = 𝑘=2 ∞ 𝑘𝑝𝑋(𝑘 − 1) = 𝑘=1 ∞ (𝑘 + 1)𝑝𝑋(𝑘) = 𝑘=1 ∞ 𝑘𝑝𝑋(𝑘) + 𝑘=1 ∞ 𝑝𝑋(𝑘) = 1 + 𝐸[𝑋] 3
Geometric Random variables 。In general,, P(X=k) PX=k X>a= P(X>a) (1-p)k-1p (1-p)k-1p 2R=a+1(1-p)k-1p 1 1 (1-p)a.1-(1-p) =(1-p)k-a-1p =px(k-a)
Geometric Random Variables • In general, 𝑃 𝑋 = 𝑘 𝑋 > 𝑎 = 𝑃(𝑋 = 𝑘) 𝑃(𝑋 > 𝑎) = (1 − 𝑝) 𝑘−1𝑝 σ𝑘=𝑎+1 ∞ (1 − 𝑝) 𝑘−1𝑝 = (1 − 𝑝) 𝑘−1𝑝 𝑝(1 − 𝑝) 𝑎∙ 1 1 − (1 − 𝑝) = (1 − 𝑝) 𝑘−𝑎−1𝑝 = 𝑝𝑋(𝑘 − 𝑎) 4
Geometric Random variables .E [fX)X>a]=E[f(X a)] ·E[f(X)IX>a] =∑fk)P(x=kIx>a)=∑fk)pxk-a) k=a+1 k=a+1 00 -〉∑fk+a)px()=Ef(X+a k三1 5
Geometric Random Variables • 𝐸 𝑓 𝑋 𝑋 > 𝑎 = 𝐸[𝑓(𝑋 + 𝑎)] • 𝐸 𝑓 𝑋 𝑋 > 𝑎 = 𝑘=𝑎+1 ∞ 𝑓(𝑘)𝑃 𝑋 = 𝑘 𝑋 > 𝑎 = 𝑘=𝑎+1 ∞ 𝑓(𝑘)𝑝𝑋(𝑘 − 𝑎) = 𝑘=1 ∞ 𝑓(𝑘 + 𝑎)𝑝𝑋(𝑘) = 𝐸[𝑓(𝑋 + 𝑎)] 5