Question

4. Families in a Faraway LandIn a faraway land,

• 20% of families have no children,
• 20% of families have exactly 1 child,
• 30% of families have exactly 2 children,
• 20% of families have exactly 3 children,
• 10% of families have exactly 4 children,
• no families have more than 4 children, and• 55% of children are boys.

1. (a) If you randomly choose a family from this faraway land, what is the expected number of children in the family? What is the expected number of daughters in the family?

2. (b) If you randomly choose a family with children from this faraway land, what is the expected number of children in the family? What is the expected number of daughters in the family?

3. (c) If you randomly choose a child from this faraway land, what is the expected number of siblings that the child has? What is the expected number of sisters that the child has?

Answer 1

Question 4

Here

Pr(0 Children) = 0.20

Pr(1 child) = 0.20

Pr(2 children) = 0.30

Pr(3 children) = 0.20

Pr(4 children) = 0.10

Pr(More than 4 children) = 0

Pr(Boys) = 0.55

Pr(Girls)= 1 - 0.55 = 0.45

Here if x is the numbe of child in one family

(a) Expected number of children in the family =E[X] =  

= 0 * 0.20 + 1 * 0.20 + 2 * 0.30 + 3 * 0.20 + 4 * 0.10 = 1.8

Expected number of daughters in the family = E[X] * Pr(Daughter) = 1.8 * 0.45 = 0.81

(b) Now it is given that we will choose a family with children from this faraway land,

so here

P(at least 1 children) =1 - Pr(No children) = 1 - 0.2 = 0.8

so here probability distribution of children in the family with atleast one children

Pr(1 children l at least one children) = 0.2/0.8 = 0.25

Pr(2 children l at least one children) = 0.3/0.8 = 0.375

Pr(3 children l at least one children) = 0.2/0.8 = 0.25

Pr(4 children l at least one children) = 0.1/0.8 = 0.125

Pr(more than one children l at least one children) = 0

Expected number of children with family having children =   1 * 0.25 + 2 * 0.375 + 3 * 0.25 + 4 * 0.125 = 2.25

Expected number of girls in the family having atleast one children = 2.25 * 0.45 = 1.0125

(c) If we randomly choose a child from this faraway land, then first we have to find the distribution of siblings

if the child is the only child, then numbers of sibling = 0

so here for x = 0 and 1, number of siblings = 0

if the child is selected from 2 children family, numbers of siblings = 1

for x = 2, number of siblings = 1

for x = 3, number of siblings = 2

for x = 4, number of siblings = 3

so here if y is the number of siblings of that choosen child

p(x = 0) = 0.25

p(x = 1) = 0.375

p(x = 2) = 0.25

p(x = 3) = 0.125

so here expected number of siblings = 0 * 0.25 + 1 * 0.375 + 2 * 0.25 + 3 * 0.125 = 1.25

expected number of sisters that the child has = 1.25 * 0.45 = 0.5625