IndependentRVs-6

# IndependentRVs-6 - Independent Discrete Variables Two...

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

1 Independent Discrete Variables Two discrete random variables X and Y are called independent if: Intuitively: knowing the value of X tells us nothing about the distribution of Y (and vice versa) If two variables are not independent, they are called dependent Similar conceptually to independent events , but we are dealing with multiple variables Keep your events and variables distinct (and clear)! y x y p x p y x p Y X , all for ) ( ) ( ) , ( Coin Flips Flip coin with probability p of “heads” Flip coin a total of n + m times Let X = number of heads in first n flips Let Y = number of heads in next m flips X and Y are independent Let Z = number of total heads in n + m flips Are X and Z independent? o What if you are told Z = 0? y m y x n x p p y m p p x n y Y x X P ) 1 ( ) 1 ( ) , ( ) ( ) ( y Y P x X P Web Server Requests Let N = # of requests to web server/day Suppose N ~ Poi( l ) Each request comes from a human (probability = p ) or from a “bot” (probability = (1 – p )), independently X = # requests from humans/day (X | N) ~ Bin(N, p ) Y = # requests from bots/day (Y | N) ~ Bin(N, 1 - p ) Note: ) ( ) | , ( ) ( ) | , ( ) , ( j i Y X P j i Y X j Y i X P j i Y X P j i Y X j Y i X P j Y i X P 0 ) | , ( j i Y X j Y i X P )! ( ) ( j i j i e j i Y X P l j i p p j i Y X j Y i X P i j i ) 1 ( ) | , ( )! ( ) 1 ( ) , ( j i i j i j i j i e p p j Y i X P Web Server Requests (cont.) Let N = # of requests to web server/day Suppose N ~ Poi( l ) Each request comes from a human (probability = p ) or from a “bot” (probability = (1 – p )), independently X = # requests from humans/day

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 ]}

### Page1 / 3

IndependentRVs-6 - Independent Discrete Variables Two...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online