A certain small town whose population consists of 100 famili

A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these children is 1 if the child is firstborn, 2 if the child is the second born, and 3 if the child is the thirdborn. (a) A random family is chosen (with equal probabilities), and then a random child within that family is chosen (with equal probabilities). Find the PMF, mean, and variance of the child’s birth rank. b.) a random child is chosen in the town (with equal prob) Find the PMF, mean, and variance of the child\'s birth rank.

Solution

Prob for each family = 1/3

Prob for each child in I family =1, ii family = 1/2, iii family =1/3

Hence child birth rank can take values as

Mean = E(X) = 1.5

Var (x) = 2.7222-2.25

= 0.4222