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)

Answer 1

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