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Unformatted text preview: 1 Whither the Binomial Recall example of sending bit string over network n = 4 bits sent over network where each bit had independent probability of corruption p = 0.1 X = number of bit corrupted. X ~ Bin(4, 0.1) In real networks, send large bit strings (length n 10 4 ) Probability of bit corruption is very small p 10-6 X ~ Bin(10 4 , 10-6 ) is unwieldy to compute Extreme n and p values arise in many cases # bit errors in file written to disk (# of typos in a book) # of elements in particular bucket of large hash table # of servers crashes in a day in giant data center # Facebook login requests that go to particular server Binomial in the Limit Recall the Binomial distribution Let l = np (equivalently: p = l / n ) When n is large, p is small, and l is moderate: Yielding: i n i p p i n i n i X P ) 1 ( )! ( ! ! ) ( i n i i i n i n n i n n n i n i n i X P i n n n ) / 1 ( ) / 1 ( ! 1 )! ( ! ! ) ( ) 1 )...( 1 ( l l l l l l l e n n ) / 1 ( 1 ) 1 )...( 1 ( i n i n n n 1 ) / 1 ( i n l l l l l e i e i i X P i i ! 1 ! 1 ) ( Poisson Random Variable X is a Poisson Random Variable: X ~ Poi( l ) X takes on values 0, 1, 2 and, for a given parameter l > 0, has distribution (PMF): Note Taylor series: So: ! ) ( i e i X P i l l 2 1 ! ... ! 2 ! 1 ! i i i e l l l l l 1 ! ! ) ( l l l l l l e e i e i e i X P i i i i i Sending Data on Network Redux Recall example of sending bit string over network Send bit string of length n = 10 4 Probability of (independent) bit corruption p = 10-6 X ~ Poi( l = 10 4 * 10-6 = 0.01) What is probability that message arrives uncorrupted? Using Y ~ Bin(10 4 , 10-6 ): 990049834 . ! ) 01 . ( ! ) ( 0 1 . e i e X P i l l 990049829 . ) ( Y P Caveat emptor: Binomial computed with built-in function in R software package, so some approximation may have occurred. Approximation are closer to you than they may appear in some software packages.are closer to you than they may appear in some software packages....
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- Spring '09