Question

A box of bulbs is inspected. The inspector draws a bulb randomly and independently, and tests it until he finds 10 defective bulbs. Suppose 5% of all bulbs are defective. Let X be the number of bulbs tested in order to find 10 defective bulbs.

• A) What is the distribution of X?

• B) What is the expected number of bulbs that must be tested in order to get 10 defective bulbs?

• C) What is the variance of X?

• D) What is the probability that 15 bulbs will be tested in order to find

10 defective bulbs?

Answer 1

A) A negative binomial experiment is a statistical experiment that has the following properties:

The experiment consists of x repeated trials.

Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.

The probability of success, denoted by $p,$ is the same on every trial.

The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

The experiment continues until $r$ successes are observed, where $r$  is specified in advance.

$X\sim NB(10 ,0.05)$

B)

$$E(X)=\dfrac{r(1-p)}{p}=\dfrac{10(1-0.05)}{0.05}=190$$

C)

$$Var(X)=\dfrac{r(1-p)}{p^2}=\dfrac{10(1-0.05)}{(0.05)^2}=3800$$

D)

$$nb(10;5,0.05)=\dbinom{15-1}{10-1}(0.05)^{10}(1-0.05)^{15-10}$$

$$=1.5\times10^{-10}$$