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Negative Binomial Distribution
This is an extension of the geometric distribution, describing the waiting time until r "ones" have
appeared. The probability of the r
th
"one" appearing on the k
th
trial is given by the negative
binomial distribution:
P (X = k) =
r1
C
k1
p
r1
(1 –p)
kr
p
in other words, the first part is the probability of r1 success in the previous k1 trails as a
binomial probability, the last tem is the probability of success.
The following is a Negative Binomial probability function with parameters (r = 6 , k= 30, p =
0.5):
A Negative Binomial Probability Function
A negative binomial distribution has:
mean = r/p and variance = r(1p)/p
2
Application:
Suppose we are at a rifle range with an old gun that misfires 5 out of 6 times.
Define ``success'' as the event the gunfires and let X be the number of failures before the third
success. Then X has a negative binomial with parameters (3, 1/6). The probability that there are
10 failures before the third success is given by:
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This note was uploaded on 10/04/2011 for the course ECO 6416 taught by Professor Staff during the Spring '08 term at University of Central Florida.
 Spring '08
 Staff

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