3.2.5 Negative Binomial Distribution - 國立臺灣大學

1 Negative Binomial DistributionIn a sequence of independent Bernoulli(p) trials, let the random variableXdenote the trialat which therthsuccess occurs, whereris a fixed integer. ThenP(X=x|r, p) =(x 1r 1)pr(1 p)x r, x=r, r+ 1, .. ,(1)and we say thatXhas a Negative Binomial (r, p) Negative Binomial Distribution is sometimes defined in terms of the random variableY=number of failures beforerth success. This formulation is statistically equivalent to theone given above in terms ofX=trial at which therth success occurs, sinceY=X r.

2 Thealternative form of the Negative Binomial Distribution isP(Y=y) =(r+y 1y)pr(1 p)y, y= 0,1, ..The Negative Binomial Distribution gets its name from the relationship(r+y 1y)= ( 1)y( ry)= ( 1)y( r)( r 1) ( r y+ 1)(y)(y 1) (2)(1),(2)which is the defining equation for Binomial coefficient with Negative integers. Along with(2), we have yP(Y=y)= 1from the Negative Binomial expansition which states that(1 +t) r= k( rk)tk= k( 1)k(r+k 1k)tk1EY= y=0y(r+y 1y)pr(1 p)y= y=1(r+y 1)!(y 1)!(r 1)!pr(1 p)y= y=1r(1 p)p(r+y 1y 1)pr+1(1 p)y 1=r(1 p)p z=0(r+ 1 +z 1z)pr+1(1 p)z=r1 similar calculation will showVarY=r(1 p) (Inverse Binomial SamplingA technique known as an inverse Binomial sampling is useful in sampling biological popula-tions.)

3 If the proportion of individuals possessing a certain characteristic ispand we sampleuntil we seersuch individuals, then the number of individuals sampled is a Negative bnomialrndom Geometric distributionThe geometric Distribution is the simplest of the waiting time distributions and is a specialcase of the Negative Binomial Distribution . Letr= 1 in (1) we haveP(X=x|p) =p(1 p)x 1, x= 1,2, .. ,which defines the pmf of a geometric random variableXwith success be interpreted as the trial at which the first success occurs, so we are waiting fora success.

4 The mean and variance ofXcan be calculated by using the Negative binomialformulas and by writingX=Y+ 1 to obtainEX=EY+ 1 =1 Pand VarX=1 geometric Distribution has an interesting property, known as the memoryless integerss > t, it is the case thatP(X > s|X > t) =P(X > s t),(3)that is, the geometric Distribution forgets what has occurred. The probability of gettingan additionals tfailures, having already observedtfailures, is the same as the probabilityof observings tfailures at the start of the establish (3), we first note that for any integern,P(X > n) =P(no success inntrials) = (1 p)n,and hence,P(X > s|X > t) =P(X > sandX > t)P(X > t)=P(X > s)P(X > t)= (1 p)s t=P(X > s t).

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