Question

In a group of 12 astronauts, 5 of them are experts in exobiology. Out of the 12 astronauts, 3 are randomly selected (without replacement) to be on the next mission. Let X be the number of experts selected for the mission:

a) Find the probability mass function of X

b) Based on your answer in a) find V ar(1.3 − 6.2X). Circle your final answer. Hint: It may be helpful to first find Var(X).

Answer 1

a)

We are given that:

Total no. of astronauts = 12

Total no. of experts in the group = 5

=> Total no. of non-experts = 12-5 = 7

No. of people to be selected from the group = 3

Now, let X denote the no. of experts in the group of 3 selected (and there are total 5 experts in the group and 7 non-experts) thus possible values of X = 0,1,2,3.

Now, the probability mass function of X is given by:

where the possible values of x = 0,1,2,3

Now,

Thus, the probability mass function of X is given by:

Now, we calculate the value of PMF for the possible values of x:

b)

First, we calculate:

Thus, the required value is given by:

For any queries, feel free to comment and ask.

If the solution was helpful to you, don't forget to upvote it by clicking on the 'thumbs up' button.