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Question
Let X be a random variable with a cumulative distribution function.
F(x) = `{{:(0",", "if" x < 0),(x/8",", "if" 0 ≤ x ≤ 1),(1/4 + x/8",", "if" 1 ≤ x ≤ 2),(3/4 + x/12",", "if" 2 ≤ x < 3),(1",", "for" 3 ≤ x):}`
Compute: (i) P(1 ≤ X ≤ 2) and (ii) P(X = 3)
Solution
W.K.T Probability density Function
f(x) = `("d"["F"(x)])/("d"x)`
f(x) = `{{:(1/8, "if" 0 ≤ x < 1),(1/8, "if" ≤ x < 2),(1/12, "if" 2 ≤ x 3),(0, "elsewhere"):}`
(i) P(1 ≤ X ≤ 2) = F(2) – F(1)
= `[3/4 + 2/12] - [1/4 + 1/8]`
= `3/4 + 1/6 - 1/4 - 1/8`
= `2/4 + 1/6 - 1/8`
= `(12 + 4 - 3)/24`
= `13/24`
(ii) P(X = 3) = 0
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