Question

In a university it is known that there are exactly the same number of men and women among faculty members. The university selects one faculty member each month for a survey on the effectiveness of new faculty mentoring program. It is understood that any faculty member can be selected more than once. Let X denote a random variable that counts the number of female faculty members selected over a period of 30 months.

1 Write down the pmf of the RV X

2 Find E(X) and σ 2 X

3 Find P(X < 2)

Answer 1

Let X denote a random variable that counts the number of female faculty members selected over a period of 30 months.

So X takes values as 0, 1, ... , 30

Because the university selects one faculty member each month for a survey on the effectiveness of new faculty mentoring program.

Here n = 30 which is fixed.

There are only two possible outcomes per trials.

Also the selection of person in one month does not affect the other month.

That is all the trials are independents.

There are equal number of male and females.

So p = probability of selection of female member = 1/2 = 0.5

The random variable X satisfies all the conditions of the binomial distribution.

So that X follows binomial distribution with n = 30 and p =0.5

The general formula of the PMF of binomial distribution is as follow:

, x = 0, 1, ..., 30

2) E(X) = n*p = 30*0.5 = 15

3  P(X < 2) = P(X = 0) + P(X = 1)

Using the formula P(X = x), we get

P(X < 2) = 0.0000000009313 + 0.0000000279397 = 0.000000028871