Question

In a box there are 5 light bulbs including 3 good bulbs and 2 broken bulbs. One person who tries to turn one by one until they get 2 good bulbs then stop. Call X is the number of times the person has tested the light bulb. Calculate the expectation, variance and standard deviation of X.
(Bonus: please give me some explanations. thanks u)

Answer 1

Given that the number of light bulbs=n=5

The number of good bulbs=3

The number of bad bulbs=2

The probability of getting good bulbs

The probability of getting bad bulbs

Here The person tries to turn on until they get 2 bulbs then stop. so it follows negative binomial distribution

Mean of the negative binomal distribution

Where r is the no.of sucess

Here the no.of sucess=r=2

Variance

Standard devation