Negative Binomial Distribution

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November 18th 2008, 10:05 PM
ccdelia7

Negative Binomial Distribution

2. Experience tells you that 5% of pumps (of a given make and model) are defective. Use the negative binomial distribution to answer the questions below.

a. What is the probability that the next pump you order is good (has no defects)? [You don’t need the negative binomial for this part.]
5 out of 100 times, the pump you will buy is defective, so, by this trend, the remaining 95 times out of 100 (on average), your pump should be fully functional. In other words, you have a 95% chance of purchasing a “good” pump.
b. What is the probability of having to order 5 pumps to get 5 good ones? I know it's .774, but how do I show this using negative binomial distribution?

c. What is the probability of having to order 6 pumps to get 5 good ones?
d. What is the probability of having to order 7 pumps to get 5 good ones?
e. How many pumps should you order to have at least a 95% chance of getting 5 good ones?
f. How many pumps should you order to have at least a 99% chance of getting 5 good ones?
November 19th 2008, 12:40 AM
mr fantastic

Quote:

Originally Posted by ccdelia7

2. Experience tells you that 5% of pumps (of a given make and model) are defective. Use the negative binomial distribution to answer the questions below.

a. What is the probability that the next pump you order is good (has no defects)? [You don’t need the negative binomial for this part.]
5 out of 100 times, the pump you will buy is defective, so, by this trend, the remaining 95 times out of 100 (on average), your pump should be fully functional. In other words, you have a 95% chance of purchasing a “good” pump.
b. What is the probability of having to order 5 pumps to get 5 good ones? I know it's .774, but how do I show this using negative binomial distribution?

c. What is the probability of having to order 6 pumps to get 5 good ones?
d. What is the probability of having to order 7 pumps to get 5 good ones?
e. How many pumps should you order to have at least a 95% chance of getting 5 good ones?
f. How many pumps should you order to have at least a 99% chance of getting 5 good ones?

Let X be the random variable number of trials before r successes.

$\Pr(X = x) = {x - 1\choose r - 1} p^r (1 - p)^{x-r} = {x - 1\choose r - 1} (0.95)^r (0.05)^{x-r}$ .

b. Substitute r = 5 and x = 5.

c. Substitute r = 5 and x = 6.

d. Substitute r = 5 and x = 7.

e. and f. require the binomial distribution:

Let Y be the random variable number of good pumps in n orders

Y ~ Binomial(n, p = 0.95).

$\Pr(Y = y) = {n \choose y} p^y (1 - p)^{n-y} \Rightarrow \Pr(Y = 5) = {n \choose 5} (0.95)^5 (0.05)^{n-5}$ .

e. Solve ${n \choose 5} (0.95)^5 (0.05)^{n-5} = 0.95$ for the smallest positive integer value of n (I suggest either trial and error or technology).

f. Solve ${n \choose 5} (0.95)^5 (0.05)^{n-5} = 0.99$ for the smallest positive integer value of n (I suggest either trial and error or technology).

And there's no answer. Perhaps the question was meant to be "How many pumps should you order to have at least a 95% chance of getting at least 5 good ones?"
November 19th 2008, 01:27 AM
ccdelia7

In parts c and d, aren't "4 choose 5" and "4 choose 6" equal to zero?
November 19th 2008, 01:34 AM
mr fantastic

Quote:

Originally Posted by ccdelia7

In parts c and d, aren't "4 choose 5" and "4 choose 6" equal to zero?

I made some mistakes in my editing of the cut and pastes I did. These mistakes have been fixed.
November 19th 2008, 02:12 AM
ccdelia7

Are you sure the equations for e and f are accurate? Using the two equations, we would have to buy less than 5 pumps to satisfy the equations themselves.
November 19th 2008, 02:22 AM
mr fantastic

Quote:

Originally Posted by ccdelia7

Are you sure the equations for e and f are accurate? Using the two equations, we would have to buy less than 5 pumps to satisfy the equations themselves.

Like I said, there's no answer if you require exactly 5 good pumps.

The questions are probably meant to be:

e. How many pumps should you order to have at least a 95% chance of getting at least 5 good ones?

f. How many pumps should you order to have at least a 99% chance of getting at least 5 good ones?

In which case:

e. You need to solve (using technology) $\Pr(Y \leq 4) \leq 0.05$ for the smallest positive integer value of n. I get n = 6.

f. You need to solve (using technology) $\Pr(Y \leq 4) \leq 0.01$ for the smallest positive integer value of n. I get n = 7.