Slide_04+--+pdf+cf+and+mgf

Slide_04+--+pdf+cf+and+mgf - The pdf, the cf, and the mgf...

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The pdf, the cf, and the mgf of a random variable • The following FT and LT relation among pdf, cf, and mgf can be utilized to find the pdf of “the sum of many independent random variables” • For discrete rvs, the “integration” is replaced by the “summation” For a given a rv , its robability density function (pdf), ( ) the characteristic function (cf), (f), and the moment generation function (mgf), (s) are connected by FT and LT as follows: (f) X X X X X fx Φ Φ Φ ± {} {} 22 ( ) () ( s ) jf X x XX sX sx X E ee f x d x F T f x Ee e f xd x L T f x ππ +∞ ⋅⋅ + −∞ + =⋅ = Φ= = ± { } {} 2 2 (f) ( ) (s) ( ) n n x X Xn n n e PX x e PX x π = = ± ±
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FACT: the pdf of “the sum of independent random variables” is the convolution of all the pdfs of the independent rvs contained in the summation. 12 N n=1 If = , with ( ) and ' are independent, t h e p d f o f () , a n d the cf of : ( ) ( ) ( ) ( ),and the mgf of : ( ) ( ) ( ) ( ) If we furthe n n n n nn X n YXX X YX X X X X X f x X s Yf y fyfy fy f f f Ys s s s ∼= Φ= Φ Φ
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Slide_04+--+pdf+cf+and+mgf - The pdf, the cf, and the mgf...

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