Advanced Topics in Probability and Stochastic Processes

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Advanced Topics in Probability and Stochastic Processes1. Linear Transformations of Normal Random Variables are also Normal()()()()()()()()()()2222222Let~,and,11The p.d.f of,exp,,0.2211The c.d.f of,exp22LetThe c.d.f of,11exp22XxXYXNa bXfxxxX FxPXxtdtYaXbY FyP YyP aXbyybPXat −=−−−  ==−−=+==+−==−−()2222211exp22Therefore,has also normal distribution with parameters,.ybatdtyabdyaaYab a−−−=−−++

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()()()()()329 5329 5329 5153227599 59 559Therefore, the temperature inalso follows normal distribution with mean 59 and SD 9.ooooooFCFCoFCF=+=+=+=+====2. Moment-generating function of the normal()()( )()()222222222Let~0,11The p.d.f of,exp2 ,.2The m.g.f of,exp1expexp221exp2exp22Since expexp2exp2exp2exp2exp2ZZXNZfzzzZ MtEtZtzzdztztdztzzztztztt−−=−−  ==−=−−−+−=−+=−+−−=−()21Sinceexp212ztdz−−−=

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