1Whither the Binomial…•Recall example of sending bit string over networkn= 4 bits sent over network where each bit had independent probability of corruption p= 0.1X = number of bit corrupted. X ~ Bin(4, 0.1)In real networks, send large bit strings (length n104)Probability of bit corruption is very small p10-6X ~ Bin(104, 10-6) is unwieldy to compute•Extreme nand pvalues 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 serverBinomial in the Limit•Recall the Binomial distribution•Let l= np(equivalently: p= l/n)•When nis large, pis small, and lis “moderate”:•Yielding:inippininiXP)1()!(!!)(iniiininninnnininiXPinnn)/1()/1(!1)!(!!)()1)...(1(lllllllenn)/1(1)1)...(1(ininnn1)/1(inllllleieiiXPii!1!1)(Poisson Random Variable•X is a PoissonRandom 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:!)(ieiXPill0210!...!2!1!0iiielllll1!!)(000lllllleeieieiXPiiiiiSending Data on Network Redux•Recall example of sending bit string over networkSend bit string of length n= 104Probability of (independent) bit corruption p= 10-6X ~ Poi(l= 104* 10-6= 0.01)What is probability that message arrives uncorrupted?Using Y ~ Bin(104, 10-6):990049834.0!0)01.0(!)0(001.0eieXPill990049829.0)0(YPCaveat 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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