Advanced Topics in Probability and Stochastic Processes

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Advanced Topics in Probability and Stochastic Processes 1. Linear Transformations of Normal Random Variables are also Normal ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 Let ~ , and , 1 1 The p.d.f of , exp , , 0. 2 2 1 1 The c.d.f of , exp 2 2 Let The c.d.f of , 1 1 exp 2 2 X x X Y X N a b X f x x x X F x P X x t dt Y aX b Y F y P Y y P aX b y y b P X a t                −    = − − −           =  = − −     = + =  = +  −   =       = − −  ( ) 2 2 2 2 2 1 1 exp 2 2 Therefore, has also normal distribution with parameters , . y b a t dt y a b dy a a Y a b a       − − −        = − − +           +    
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( ) ( ) ( ) ( ) ( ) 32 9 5 32 9 5 32 9 5 15 32 27 59 9 5 9 5 5 9 Therefore, the temperature in also follows normal distribution with mean 59 and SD 9. o o o o o o F C F C o F C F     = + = + = +  = + = = =  = 2. Moment - generating function of the normal ( ) ( )   ( )         ( )           ( )     2 2 2 2 2 2 2 2 2 Let ~ 0,1 1 The p.d.f of , exp 2 , . 2 The m.g.f of , exp 1 exp exp 2 2 1 exp 2 exp 2 2 Since exp exp 2 exp 2 exp 2 exp 2 exp 2 Z Z X N Z f z z z Z M t E tZ tz z dz t z t dz tz z z tz t z t t     −  −  = − −    =     = − = − − −   + − = − +     = − + − −   = −   ( )   2 1 Since exp 2 1 2 z t dz   −   − − =     

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