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

# Slide_04+--+pdf+cf+and+mgf - The pdf the cf and the mgf of...

This preview shows pages 1–3. Sign up to view the full content.

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 = = ± ±

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online