A continuous random variable X has probability density function f(x) = 0.1x - 0.3 for 4 ≤ x ≤ 6, f(x) = 0.3 for 6 ≤ x ≤ 8, and f(x) = 0 otherwise. Find P(5 ≤ X ≤ 7).

A continuous random variable $X$ has probability density function $f(x) = 0.1x - 0.3$ for $4 \leq x \leq 6$ , $f(x) = 0.3$ for $6 \leq x \leq 8$ , and $f(x) = 0$ otherwise. Find $P(5 \leq X \leq 7)$ .

Model answer

0.55

Also accepted: 0,55, 11/20

Explanation

P(5 \leq X \leq 7) = \int_5^6 (0.1x-0.3)\,dx + \int_6^7 0.3\,dx

. The first part is

0.05(6^2-5^2) - 0.3 = 0.55 - 0.3 = 0.25

and the second is

0.3

, giving

0.25 + 0.3 = 0.55

Derived from ZIMSEC Statistics 6046/2 Paper 2, Specimen Paper, Q1

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