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 (i.e. Fill out your final answers in the table):

x
P (X = 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) The hypergeometric distribution is used to calculate probabilities when sampling without replacement.The pmf of hypergeometric distribution is given by :

where

K is the number of "successes" in the population
X is the number of "successes" in the sample
N is the size of the population
n is the number sampled

We have given,

Group of astronauts, N=12

Experts in exobiology, k = 5

Sample size i.e. total astronauts out of 12 selected for mission , n =3

X: the number of experts selected for the mission.

Now,

b) The variance of hypergeometric distribitution is given by:

                  

                   

Now,