The Negative Binomial Distribution

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The Negative Binomial Distribution

• Often we want to wait for an event to occur. We
  may wait for an accident to occur at a certain
  intersection or families may wait until a male
  child is born. Such events are considered in the
  geometric random variable.

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• production processes items not meeting production
  specifications occasionally occur, but the
  detection of one such item may not in itself be a
  cause for alarm. Instead we may wait for a
  certain number of events to occur. The random
  variable involved is called a negative binomial
  random variable. We consider variables here.

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• One of two events (usually denoted by "success"
  or "failure") occur at each trial of the
  experiment,
• The trials are independent,
• The probability of success at any trial is p
  which we denote by P(S) p, and so the
  probability of failure at any trial is P(F) 1 -
  p q. We assume that these probabilities remain
  constant throughout our series of trials.

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• In the binomial random variable, we have a fixed
  number of trials, say n, and the random variable
  X is the number of successes that occur.
• In Geometric random variable, we wait for the
  occurrence of the single event, so X is the
  waiting time for the event to occur.

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negative binomial random variable

• In the negative binomial random variable, we wait
  until a given number of events have occurred
  (usually more the one).
• For example, lets wait until two events have
  occurred. Let X be the total number of trials
  necessary. The sample space and the associated
  probabilities now as follows

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Negative binomial
X Points Probability
2 SS p2
3 FSS or SFS 2 p2q
4 FFSS or SFFS or FSFS 3p2q2
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Negative binomial

• The pattern in the probabilities here continues
  and in this case it is not difficult to guess
  what that pattern is. For example, if we want the
  second event to occur at the fifth trial (X5),
  the sample points are
• FFFSS FSFFS SFFFS FFSFS
• Since each of these sequences has the
  probability p2q3,we see that
• P(X5) 4p2q3.

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Negative binomial

• We can determine this formula without writing
  down all the sample points in this way. We know
  that the fifth trial is the second S, so the
  first four trials must contain exactly one S and
  three Fs. These four trials constitute a
  binomial sequence, that is, we have four trials
  and want the probability of exactly one success
  among them. This has probability

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Negative binomial

• Now we put a success, with probability p, at the
  fifth trial so

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Negative binomial

• Lets generalize this reasoning. If the second
  success is to occur at trial x, the first x-1
  trials must contain exactly one S and x-2 Fs.
  This has probability

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• Now, putting a success at the xth trial, we have

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Negative binomial

• and we have determined the pattern for the
  waiting time for the second success in a series
  of trials that meet the conditions we gave.
• Before proceeding, we ought to determine if we
  have assigned a total probability of 1 to the
  sample space.  So we must check that

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• To see that this is in fact correct, let S be the
  right hand side of the expression above

multiplying through by q we have
Subtracting the second series from the first
gives
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• and we recognize that
• is a geometric series with the first term 1
  and ratio q so

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• It follows that
• so that
• pS p
• meaning that S 1.