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...

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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 the ?rstborn, 2 if the child is the secondborn, 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 probabilities). Find the PMF, mean, and variance of the child’s birth rank.

Answered Same DayDec 26, 2021

Answer To: A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50...

Robert answered on Dec 26 2021

128 Votes

a)
Let X be the birth rank of the chosen child. Then by the Law of Total Probability(LOTP),
?(? = ?) = ∑ ?(? = ?|?). ?(?)3?=1 ?ℎ??? ? = 1,2,3
T=Family type indicates number of children in a family can be Type1, Type2, and Type 3 with
probabilities 30% , 50%, 20% respectively. ?(? = ?|?) would of course be 1, ½ or 1/3
depending upon T=1, 2 or 3
PMF ( Probability Mass Function)
?(? = 1) =
30
100
. 1...

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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...

Answer To: A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50...

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