IndependentRVs-4

IndependentRVs-4 - Independent Discrete Variables Two...

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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
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IndependentRVs-4 - Independent Discrete Variables Two...

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