Tutorial 4: Discrete RandomVariables 2BaoxiangWangbxwang@cseSpring2017
Tutorial 4: Discrete Random Variables 2 Baoxiang Wang bxwang@cse Spring 2017 1
Geometric Random Variables.ThePMFofageometricrandomvariableispx(k) = (1 -p)k-1p: E[X|X > 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? For k > 1,(1-p)k-1p(1-p)k-1pP(X=k)P(X = k|X > 1)ZK=2(1-p)k-1pP(X>1)p(1-p):1-(1-p)= (1 -p)k-2p = px(k - 1)· E[XIX > 1] = Z=2 kP(X = k]X > 1)880kpx(k - 1)> (k +1)px(k)kpx(k) +px(k)k=1k=1k=2k=1 i+E[X]
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)P(X = k|X > a)P(X >a)(1 -p)k-1p-1p1ZK=a+1(1 -p)k-1pp(1-p)a:1-(1-p)=(1 -p)k-a-1pr= 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[f(X)IX > a] = E[f(X +a)]· E[f(X)|X > a]8ZZf(k)P(X = k|X > a)f(k)px(k-a)/k=a+1k=a+1807f(k + a)px(k) = E[f(X +a))一k=1
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