PROBABILITY
i=1, 2 ,...,N, is distributed asXi∼Bin(ni,p)thenZ=X 1 +X 2 +···+XNis
distributed asZ∼Bin(n 1 +n 2 +···+nN,p), as would be expected since the result
of
∑
initrials cannot depend on how they are split up. A similar proof is also
possible using either the probability or cumulant generating functions.
Unfortunately, no equivalent simple result exists for the probability distributionof thedifferenceZ=X−Yof two binomially distributed variables.
30.8.2 The geometric and negative binomial distributionsA special case of the binomial distribution occurs when instead of the number of
successes we consider the discrete random variable
X= number of trials required to obtain the first success.The probability thatxtrials are required in order to obtain the first success, is
simply the probability of obtainingx−1 failures followed by one success. If the
probability of a success on each trial isp,thenforx> 0
f(x)=Pr(X=x)=(1−p)x−^1 p=qx−^1 p,whereq=1−p. This distribution is sometimes called thegeometric distribution.
The probability generating function for this distribution is given in (30.78). By
replacingtbyetin (30.78) we immediately obtain the MGF of the geometric
distribution
M(t)=pet
1 −qet,from which its mean and variance are found to be
E[X]=1
p,V[X]=q
p^2.Another distribution closely related to the binomial is the negative binomialdistribution. This describes the probability distribution of the random variable
X= number of failures before therth success.One way of obtainingxfailures before therth success is to haver−1 successes
followed byxfailures followed by therth success, for which the probability is
pp···p
︸︷︷︸
r−1times×qq···q
︸ ︷︷︸
xtimes×p=prqx.However, the firstr+x−1 factors constitute just one permutation ofr− 1
successes andxfailures. The total number of permutations of theser+x− 1
objects, of whichr−1 are identical and of type 1 andxare identical and of type