Chapter 9 Sum of Random Variables - Korea...
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Chapter 9 Sum of Random Variables Chang-Su Kim Korea 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.
Transcript of Chapter 9 Sum of Random Variables - Korea...
Chapter 9Sum of Random Variables
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.
Random Variable PMF or PDF MGF 𝛷𝛷𝑋𝑋(𝑠𝑠)
Bernoulli (𝑝𝑝) 𝑃𝑃𝑋𝑋 𝑥𝑥 = �1 − 𝑝𝑝 𝑥𝑥 = 0𝑝𝑝 𝑥𝑥 = 1 1 − 𝑝𝑝 + 𝑝𝑝𝑒𝑒𝑠𝑠
Binomial (𝑛𝑛, 𝑝𝑝) 𝑃𝑃𝑋𝑋 𝑥𝑥 = 𝑛𝑛𝑥𝑥 𝑝𝑝𝑥𝑥 1 − 𝑝𝑝 𝑥𝑥 1 − 𝑝𝑝 + 𝑝𝑝𝑒𝑒𝑠𝑠 𝑛𝑛
Poisson (𝛼𝛼) 𝑃𝑃𝑋𝑋 𝑥𝑥 =𝛼𝛼𝑥𝑥𝑒𝑒−𝛼𝛼
𝑥𝑥!𝑒𝑒𝛼𝛼(𝑒𝑒𝑠𝑠−1)
Exponential 𝜆𝜆 𝑓𝑓𝑋𝑋 𝑥𝑥 = 𝜆𝜆𝑒𝑒−𝜆𝜆𝑥𝑥𝜆𝜆
𝜆𝜆 − 𝑠𝑠
Erlang 𝑛𝑛, 𝜆𝜆 𝑓𝑓𝑋𝑋 𝑥𝑥 =𝜆𝜆𝑛𝑛𝑥𝑥𝑛𝑛−1𝑒𝑒−𝜆𝜆𝑥𝑥
𝑛𝑛 − 1 !𝜆𝜆
𝜆𝜆 − 𝑠𝑠
𝑛𝑛
Gaussian (𝜇𝜇,𝜎𝜎) 𝑓𝑓𝑋𝑋 𝑥𝑥 =1
𝜎𝜎 2𝜋𝜋𝑒𝑒−
𝑥𝑥−𝜇𝜇 2
2𝜎𝜎2 𝑒𝑒𝑠𝑠𝜇𝜇+𝑠𝑠2𝜎𝜎22
