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The geometric distribution - University of Utah

The geometric distributionSo far, we have seen only examples of random variables that have a finitenumber of possible values. However, our rules of probability allow us toalso study random variables that have a countable [but possibly infinite]number of possible values. The word countable means that you can labelthe possible values as 1,2,.. A theorem of Cantor states that the numberof elements of the real line is uncountable. And so is the number ofelements of any nonempty closed/open interval. Therefore, the conditionthat a random variableXhas a countable number of possible values is say thatXhas thegeometric distributionwith parameter!:= 1 "ifP{X=#}="# 1!(#=1$2$%%%)%We have seen this distribution before. Here is one way it arises naturally:Suppose we toss a!-coin [ ,P(heads)=!,P(tails)="=1 !] untilthe first heads arrives. IfXdenotes the number of tosses, thenXhas theGeometric(!)

The Poisson distribution 57 The negative binomial distribution The negative binomial distribution is a generalization of the geometric [and not the binomial, as the name might suggest]. Let us fix an integer) ≥ 1; then we toss a!-coin until the)th heads occur. Let X) denote the total number of tosses. Example 4 (The negative binomial ...

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