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MoreDiscreteDist-4

MoreDiscreteDist-4 - Whither the Binomial Recall example of...

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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 0 2 1 0 ! ... ! 2 ! 1 ! 0 i i i e l l l l l 1 ! ! ) ( 0 0 0 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 . 0 ! 0 ) 01 . 0 ( ! ) 0 ( 0 01 . 0 e i e X P i l l 990049829 . 0 ) 0 ( 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.
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