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三1Geometric 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