{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MoreIndRVs+ConditionalDist-6

# MoreIndRVs+ConditionalDist-6 - Sum of Independent Binomial...

This preview shows pages 1–2. Sign up to view the full content.

1 Sum of Independent Binomial RVs Let X and Y be independent random variables X ~ Bin(n 1 , p) and Y ~ Bin(n 2 , p) X + Y ~ Bin(n 1 + n 2 , p) Intuition: X has n 1 trials and Y has n 2 trials o Each trial has same “success” probability p Define Z to be n 1 + n 2 trials, each with success prob. p Z ~ Bin(n 1 + n 2 , p), and also Z = X + Y More generally: X i ~ Bin(n i , p) for 1 i N p n X N i i n i i , Bin ~ 1 1 Sum of Independent Poisson RVs Let X and Y be independent random variables X ~ Poi( l 1 ) and Y ~ Poi( l 2 ) X + Y ~ Poi( l 1 + l 2 ) Proof: (just for reference) Rewrite (X + Y = n ) as (X = k , Y = n k ) where 0 k n Noting Binomial theorem: so, X + Y = n ~ Poi( l 1 + l 2 ) n k n k k n Y P k X P k n Y k X P n Y X P 0 0 ) ( ) ( ) , ( ) ( n k k n k n k k n k n k k n k k n k n n e k n k e k n e k e 0 2 1 ) ( 0 2 1 ) ( 0 2 1 )! ( ! ! ! )! ( ! )! ( ! 2 1 2 1 2 1 l l l l l l l l l l l l n n e n Y X P 2 1 ) ( ! ) ( 2 1 l l l l n k k n k n k n k n 0 2 1 2 1 )! ( ! ! ) ( l l l l Reference: Sum of Independent RVs Let X and Y be independent Binomial RVs X ~ Bin(n 1 , p) and Y ~ Bin(n 2 , p) X + Y ~ Bin(n 1 + n 2 , p) More generally, let X i ~ Bin(n i , p) for 1 ≤ i ≤ N, then Let X and Y be independent Poisson RVs X ~ Poi( l 1 ) and Y ~ Poi( l 2 ) X + Y ~ Poi( l 1 + l 2 ) More generally, let X i ~ Poi( l i ) for 1 ≤ i ≤ N, then p n X N i i N i i , Bin ~ 1 1 N i i N i i X 1 1 Poi ~ l Expected Values of Sums Let g(X, Y) = X + Y. Compute E[g(X, Y)] = E[X + Y] E[X + Y] = E[X] + E[Y] Generalized: Holds regardless of dependency between X i ’s We’ll prove this next time n i i n i i X E X E 1 1 ] [ Dance, Dance, Convolution Let X and Y be independent random variables Cumulative Distribution Function (CDF) of X + Y: F X+Y is called convolution of F X and F Y Probability Density Function (PDF) of X + Y, analogous: In discrete case, replace with , and f ( y ) with p ( y ) ) ( ) ( a Y X P a F Y X    y y a x Y X a y x Y X dy y f dx x f dy dx y f x f ) ( ) ( ) ( ) (  y Y X dy y f y a F ) ( ) (  y Y X Y X dy y f y a f a f ) ( ) ( ) (  y y Sum of Independent Uniform RVs Let X and Y be independent random variables X ~ Uni(0, 1) and Y ~ Uni(0, 1) f ( a ) = 1 for 0 a 1 What is PDF of X + Y?

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

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

{[ snackBarMessage ]}