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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
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Solution
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):}`
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