Chapter 7Conditional Probability Models

Chang-Su KimKorea University

These notes are modified from the files, provided by R. D. Yates and D. J. Goodman who are the authors of the textbook “Probability and Stochastic Processes,” and can be used only for the class KECE209(03) in Korea University.

Terminology Using a Dice Example

Sample space: 𝑆𝑆𝑋𝑋 = 1, 2, 3, 4, 5, 6Event: 𝐵𝐵 = 2, 4, 6 ⊂ 𝑆𝑆𝑋𝑋Conditional Probability: 𝑃𝑃𝑋𝑋|𝐵𝐵 4 = P[𝑋𝑋 = 4|𝐵𝐵]

= probability that outcome 4 occurs given that B happens

Caution: In this example, 𝐻𝐻 and 𝑁𝑁 are not subsets of 𝑆𝑆𝑋𝑋

Caution: In this example, 𝐵𝐵0 and 𝐵𝐵1 are not subsets of 𝑆𝑆𝑋𝑋

𝑃𝑃𝑋𝑋,𝑌𝑌|𝐵𝐵 𝑥𝑥,𝑦𝑦 : conditional PMF𝑓𝑓𝑋𝑋,𝑌𝑌|𝐵𝐵 𝑥𝑥,𝑦𝑦 : conditional PDF𝐸𝐸[𝑔𝑔(𝑋𝑋,𝑌𝑌)|𝐵𝐵]: conditional expectation

𝐸𝐸 𝐸𝐸 𝑋𝑋 𝑌𝑌 = 𝐸𝐸 𝑔𝑔 𝑌𝑌 = �𝑦𝑦

𝑔𝑔 𝑦𝑦 𝑃𝑃𝑌𝑌(𝑦𝑦)

Optional

About correlation coefficient 𝝆𝝆𝑿𝑿,𝒀𝒀