Question

Find the probability mass function and cumulative distribution function of a number of girl children in families with 4 children, assuming equal probabilities for boys and girls

Answer 1

Let X be the random variable denotes the number of girl child among 4 children

X = {0, 1, 2, 3, 4}

Values of the random variable	0	1	2	3	4	Total
Number of elements in inverse image	1	4	6	4	1	16

(i) Probability mass function

x	0	1	2	3	4	Total
f(x)	`1/16`	`4/16`	`6/16`	`4/16`	`1/16`	1

(ii) Cumulative distribution

F(x) = P(X ≤ x)

= `sum_(x_"i" ≤ x) "P"("X" = x_"i")` 

P(X < 0) = 0 for `- oo < x < 0`

F(0) = P(X ≤ 0) 

= `P(X = 0)

= `1/16`

F(1) = P(X ≤ 1) = P(X = 0) + P(X = 1)`

= `1/16 + 4/16`

= `5/16`

F(2) = P(X ≤ 2)

= P(X = 0) + P(X = 1) + P(X = 2)

= `1/16 + 4/16 + 6/16`

= `11/16`

F(3) = P(X ≤ 3)

= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

= `1/16 + 4/16 + 6/16 + 4/16`

= `15/16`

F(4) = P(X ≤ 4)

= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

= `15/16 + 1/16`

 = 1

F(x) = `{{:(0",",  "For"  x < 0),(1/16",",  "For"  x ≤ 0),(5/16",",  "For"   x ≤ 1), (11/16",",  "For"  x ≤ 2),(15/16",",  "For"  x ≤ 3),(1",",  "For"  x ≤ 4):}`