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!"#$%& (")*"+,-. ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 0

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The negative binomial rv and distribution are based on an experiment satisfying the following

conditions:

1. The experiment consists of a sequence of independent trials.2. Each trial can result in either a success (S) or a failure (F).3. The probability of success is constant from trial to trial, soP(S on trial i) 5p for i 51, 2, 3, . . . .4. The experiment continues (trials are performed) until a total of r successes have

been observed, where r is a specified positive integer.

The random variable of interest isX = the number of trials that precede the rth success;X iscalled a negative binomial random variable because, in contrast to the binomial rv, the number

of successes is fixed and the number of trials is random.

The pmf of the negative binomial rv X with parameters r = number of Ss and p = P(S) is

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!"#$%&' )#*%+#,-%./ 01$/"2 31+#1,&.*

BC% 4*.7)-2- !"#$%& (")*"+,-.The probability mass function (pmf) of a single discrete rv X specifies how much probability mass is

placed on each possibleX value. The joint pmf of two discrete rvsX and Y describes how much

probability mass is placed on each possible pair of values (x, y).

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The marginal probability mass function ofX, denoted by!!!!!, is given by

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Similarly, the marginal probability mass function of Yis

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I#$-A-#$-#2 !"#$%& (")*"+,-.The definition says that two variables are independent if their joint pmf or pdf is the product of

the two marginal pmfs or pdfs. Intuitively, independence says that knowing the value of one of

the variables does not provide additional information about what the value of the other variable

might be.

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!-,"2*9- ?66*7*-#73If #1 and #2 denote two unbiased estimators for the same parameter #, we prefer to use the

estimator with the smaller variance.

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hypothesis should be rejected. Thep-value is the smallest value of &for which thenull hypothesis can be rejected. Thus, if the desired value of &is greater than orequal to thep-value, the null hypothesis is rejected for that value of &.

Indeed, the null hypothesis should be rejected for any value of &down to andincluding thep-value.

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