# Let X be amount of time for which a book is taken out of library by randomly selected student and suppose X has p.d.f f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise. Calculate: P(0.5 ≤ x ≤ 1.5) - Mathematics and Statistics

Sum

Let X be amount of time for which a book is taken out of library by randomly selected student and suppose X has p.d.f

f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.

Calculate: P(0.5 ≤ x ≤ 1.5)

#### Solution

f(x) = 0.5 x ,      0 ≤ x ≤ 2

= 0 ,            otherwise

P (0.5 ≤ X ≤ 1.5)

= int_0.5^1.5 0.5  "x"

= 0.5  int_0.5^1.5 "x dx"

= 1/2 xx ["x"^2/2]_0.5^1.5

= 1/2 xx 1/2 [(1.5)^2 - (0.5)^2]

= 1/2 xx 1/2 [2.25 - 0.25]

= 1/4 xx 2 = 1/2

P (0.5 ≤ X ≤ 1.5) = 1/2

Concept: Probability Distribution of Discrete Random Variables
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