Tutorial 4:Discrete Random Variables 2 Baoxiang Wang bxwang@cse Spring 2017 1
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 ·E[XX>1]=1+E[X] 2
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 For k 1, P(X=klX >1)= P(X=k 2=1-p)k-1 2=_(1-p)k-1卫 (X>1) 2R2-2(1-p)k-1p 1 p(1-p)1-1-p =(1-p)k-2p=px(k-1) ·EXX>1]=∑=2k(X=kX>1) -夏p=+1p肉公n.+的 k=2 k三1 =1+E[X] 3
Geometric Random Variables • For 𝑘 > 1, 𝑃 𝑋 = 𝑘 𝑋 > 1 = 𝑃(𝑋=𝑘) 𝑃(𝑋>1) = (1−𝑝) 𝑘−1𝑝 (1−𝑝) ∞ 𝑘−1𝑝 𝑘=2 = (1−𝑝) 𝑘−1𝑝 𝑝(1−𝑝)∙ 1 1−(1−𝑝) = (1 − 𝑝) 𝑘−2𝑝 = 𝑝𝑋(𝑘 − 1) • 𝐸 𝑋 𝑋 > 1 = 𝑘𝑃 𝑋 = 𝑘 𝑋 > 1 ∞ 𝑘=2 = 𝑘𝑝𝑋(𝑘 − 1) ∞ 𝑘=2 = (𝑘 + 1)𝑝𝑋(𝑘) ∞ 𝑘=1 = 𝑘𝑝𝑋(𝑘) ∞ 𝑘=1 + 𝑝𝑋(𝑘) ∞ 𝑘=1 = 1 + 𝐸[𝑋] 3
Geometric Random variables ·In general, P=k) P(X=k X>a)= P(X>a) (1-p)k-1p (1-p)k-1p 28=a+1(1-p)k-1p 1 1 (1-p)a.T-1-p) =(1-p)k-a-1p =px(k-a) 4
Geometric Random Variables • In general, 𝑃 𝑋 = 𝑘 𝑋 > 𝑎 = 𝑃(𝑋 = 𝑘) 𝑃(𝑋 > 𝑎) = (1 − 𝑝) 𝑘−1𝑝 (1 − 𝑝) 𝑘−1𝑝 ∞ 𝑘=𝑎+1 = (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[f(X)X a]=E[f(X +a)] ·E[f(X)IX>a] 00 =∑f)P(X=kX>a)=∑f)px(k-) k=a+1 k=a+1 00 =f(k+a)px(k)=EIf(X+a)] k三1
Geometric Random Variables • 𝐸 𝑓 𝑋 𝑋 > 𝑎 = 𝐸[𝑓(𝑋 + 𝑎)] • 𝐸 𝑓 𝑋 𝑋 > 𝑎 = 𝑓(𝑘)𝑃 𝑋 = 𝑘 𝑋 > 𝑎 ∞ 𝑘=𝑎+1 = 𝑓(𝑘)𝑝𝑋(𝑘 − 𝑎) ∞ 𝑘=𝑎+1 = 𝑓(𝑘 + 𝑎)𝑝𝑋(𝑘) ∞ 𝑘=1 = 𝐸[𝑓(𝑋 + 𝑎)] 5