The Poisson Distribution

I Many experiments consist of observing the occurrence times of random arrivals. I Examples include arrivals of customers for service, arrivals of calls at a switchboard, occurrences of ﬂoods and other natural and man-made disasters, and so forth. I The family of Poisson distributions is used to model the number of such arrivals that occur in a ﬁxed time period. I Poisson distributions are also useful approximations to binomial distributions with very small success probabilities. The Poisson Distribution Let > 0. A random variable X has a Poisson distribution with parameter if its pmf is as follows

x e if x =0, 1, 2,... f (x)= x! (0 otherwise

... 1,2 , . XELO , ) Pr(×=x)=fp4=eȾ ,M÷

1>0 Izo 's r

' ' ' ' . . . .=e It . . .at IEIT e It ,÷ .x+¥ 's 't. The Poisson Distribution

Find the meand. and the variance of the Poisson distribution. '

. . . No ,M÷ . ftp.e x=o,m ,

" . .ee Ⱦ , EW=€ .tw II. 's'÷II,e at÷

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Elxcx - D= El Ex )

' '

' = . . . - - = e ' D ) crh , = INK fin In It Eeii IT

= et N . II. M÷ = ii

= = I = E ( xtx ) Eat ) It El *) xttx

- - Var ( X ) = EH ) EIDE Mti N = x The Poisson Distribution Find the mgf of the Poisson distribution. . th × ' ' et " et . YH ) = E ( et = EE fix ) et y÷ ) 5¥.

" ' ' let 't = . et e II. k¥⇐0 et ex The Poisson Distribution

If the random variables X1,...,Xk are independent and if Xi has the Poisson distribution with mean i (i =1,...,k),then the sum X + ...+ X has the Poisson distribution with mean + + . 1 k 1 ··· k

Find the mgf of XitXzt . . .+Xk

= the . rv of mgfs ( mgf of a sum of indep product )

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, ) . . .e^u(e± .

. e mgt of XH . .tl/e=e'

' ' the . . ' ' the ) ( et ) = [

Poisson r.ir . with = mgf of a

. +

. . param Xithz , the The Negative Binomial Distribution

I Earlier we learned that, in n Bernoulli trials with probability of success p, the number of successes has the binomial distribution with parameters n and p. I Instead of counting successes in a fixed number of trials, it is often necessary to observe the trials until we see a fixed number of successes. I For example, while monitoring a piece of equipment to see when it needs maintenance, we might let it run until it produces a fixed number of errors and then repair it. I The number of failures until a fixed number of successes has a distribution in the family of negative binomial distributions. The Negative Binomial Distribution Suppose that an infinite sequence of Bernoulli trials with probability of success p are available. Let X be the number of failures that occur before rth success. Find the pmf of X .(Thisdefinesthenegative binomial distribution.)

at leastr times . :experiment is performed X= # of failures before eels successes

11=0 ( the first r trials are all successes )

' last trial a * hi trials < ftp.hdss ( . musty.by )

last trials a- x ( rex f "rIs[ malamute! )

' ' " Pr(X=x) "I=P)pr( i . pl w ' ' " Pr ( X=x) "I=P)pr( , .pl The Geometric distribution The most common special case of a negative binomial random variable is one for which r =1. This would be the number of failures until the ﬁrst success.

I 2 . . E . X LO , , , }

- Pr x = i ( X= ) p ( PY The Negative Binomial Distribution

Let X1, X2,...,Xr be iid such that each Xi has a geometric distribution with parameter p.Then,thesum

X + X + + X 1 2 ··· r has a negative binomial distribution with parameters r, p. The Negative Binomial Distribution Find the mgf of the Negative Binomial distribution. The Begative Binomial Distribution Find the mean and the variance of the Negative Binomial distribution with parameters r, p.