Chapter 13 The Poisson Distribution Jeanne Antoinette Poisson 1721 1764 Marquise de Pompadour was a member of the French court and was the official chief mistress of Louis XV from 1745 until her death The pompadour hairstyle was named for her In addition poisson is French for fish The Poisson distribution however is named for Simeon Denis Poisson 1781 1840 a French mathematician geometer and physicist 13 1 Specification of the Poisson Distribution In this chapter we will study a family of probability distributions for a countably infinite sample space each member of which is called a Poisson distribution Recall that a binomial distribution is characterized by the values of two parameters n and p A Poisson distribution is simpler in that it has only one parameter which we denote by pronounced theta Many books and websites use pronounced lambda instead of We save for a related purpose The parameter must be positive 0 Below is the formula for computing probabilities for the Poisson P X x e x for x 0 1 2 3 x In this equation e is the famous number from calculus e n lim 1 1 n n 2 71828 You might recall from the study of infinite series in calculus that X bx x eb x 0 for any real number b Thus X x 0 P X x e X x 0 321 x x e e 1 13 1 Table 13 1 A comparison of three probability distributions Distribution of X is Poisson 1 Bin 1000 0 001 Bin 500 0 002 Mean 1 1 1 Variance 1 0 999 0 998 x P X x P X x P X x 0 0 3679 0 3677 0 3675 1 0 3679 0 3681 0 3682 2 0 1839 0 1840 0 1841 3 0 0613 0 0613 0 0613 4 0 0153 0 0153 0 0153 5 0 0031 0 0030 0 0030 6 0 0005 0 0005 0 0005 0 0001 0 0001 0 0001 7 Total 1 0000 1 0000 1 0000 Thus we see that Formula 13 1 is a mathematically valid way to assign probabilities to the nonnegative integers i e all probabilities are nonnegative indeed they are positive and they sum to one The mean of the Poisson is its parameter i e This can be proven using calculus and a similar argument shows that the variance of a Poisson is also equal to i e 2 and When I write X Poisson I mean that X is a random variable with its probability distribution given by the Poisson distribution with parameter value I ask you for patience I am going to delay my explanation of why the Poisson distribution is important in science As we will see the Poisson distribution is closely tied to the binomial For example let s spend a few minutes looking at the three probability distributions presented in Table 13 1 There is a wealth of useful information in this table In particular 1 If you were distressed that a Poisson random variable has an infinite number of possible values namely every nonnegative integer agonize no longer We see from the table that for 1 99 99 of the Poisson probability is assigned to the event X 6 2 If you read down the three columns of probabilities you will see that the entries are nearly identical Certainly any one column of probabilities provides good approximations to the entries in any other column Thus in some situations a Poisson distribution can be used as an approximation to a binomial distribution 3 What do we need for the Poisson to be a good approximation to a binomial First we need to have the means of the distributions match i e we need to use the Poisson with np as I did in Table 13 1 The variance of a binomial npq is necessarily smaller than the mean 322 np because q 1 Thus the variance of a binomial cannot be made to match the variance of the Poisson Variance of binomial npq np variance of Poisson If however p is very close to 0 then q is very close to one and the variances almost match as illustrated in Table 13 1 I will summarize the above observations in the following result Result 13 1 The Poisson approximation to the binomial The Bin n p distribution can be wellapproximated by the Poisson distribution if the following conditions are met 1 The distributions have the same mean i e np 2 The value of n is large and p is close to zero In particular the variance of the binomial npq should be very close to the variance of the Poisson np As a practical matter we mostly use this result if n 1 000 because we can easily obtain exact binomial probabilities from a website for n 1 000 Also if np 25 our general guideline from Chapter 11 states that we may use a Normal curve to obtain a good approximation to the binomial Thus again as a practical matter we mostly use this result if np 25 allowing us some indecision as to which approximation to use at np 25 Normal or Poisson Poisson probabilities can be computed by hand with a scientific calculator Alternatively the following website can be used http stattrek com Tables Poisson aspx I will give an example to illustrate the use of this site Let X Poisson The website calculates five probabilities for you P X x P X x P X x P X x and P X x You must give as input your value of and a value of x Suppose that I have X Poisson 10 and I am interested in P X 8 I go to the site and enter 8 in the box Poisson random variable and I enter 10 in the box Average rate of success I click on the Calculate box and the site gives me the following answers P X 8 0 1126 P X 8 0 2202 P X 8 0 3328 P X 8 0 6672 and P X 8 0 7798 As with our binomial calculator there is a great deal of redundancy in these five answers 323 13 1 1 The Normal Approximation to the Poisson Please look at the Poisson 1 probabilities in Table 13 1 We see that P X 0 P X 1 and as x increases beyond 1 P X x decreases Thus without actually drawing the probability histogram of the Poisson 1 we know that it is strongly skewed to the right indeed it has no left tail For 1 the probability histogram is even more skewed than it is for our tabled 1 As the value of increases the amount of skewness in the probability histogram decreases but the Poisson is never perfectly symmetric In this course I advocate the general guideline that if 25 then the Poisson s probability histogram is approximately symmetric and bell shaped One can quibble about my choice of 25 and I wouldn t argue about it much This last statement suggests that we might use a Normal curve to compute …
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