In a group of five items two are defective Find the distribu

In a group of five items, two are defective. Find the distribution of N, the number of draws we need to find the first defective item. Find the mean and variance of N.

Solution

We have given that group of 5 items.

In that 5 items two are defective.

Let us consider random variable N is number of draws we need to find the first defective item.

Thus N follows Geometric distribution with parameter p.

The features of Geometric distribution are,

1) Repeated binomial trials.

2) Continue until first success.

3) Find probability that first success comes on nth trial.

4) Probability of success on each trial is p.

The probability mass function of N is,

P(N) = p * q^N  

Here there are two cases with replacement or without replacement.

Here we consider the case with replace.

p = P(defective items) = number of defective items / total number of items

= 2 / 5

q = 1 - p = 3 / 5

P(one draw) = p = 2 / 5

P(two draws) = pq = 3 / 5 * 2 / 5

P(three draws) = pq^2 = 2 / 5 * ( 3 / 5)^2

Mean of N = 1 / p = 1 / (2/5) = 5 / 2 = 2.5

variance of N = q / p^2 = (3 / 5) / (2 /5)^2 = 3.75