A continuous random variable X has probability density function f(x) = 0.1x + k for 4 ≤ x ≤ 6, f(x) = 0.3 for 6 ≤ x ≤ 8, and f(x) = 0 otherwise. Find the value of the constant k.

A continuous random variable $X$ has probability density function $f(x) = 0.1x + k$ for $4 \leq x \leq 6$ , $f(x) = 0.3$ for $6 \leq x \leq 8$ , and $f(x) = 0$ otherwise. Find the value of the constant $k$ .

Model answer

-0.3

Also accepted: -0,3, k=-0.3

Explanation

The density must integrate to 1:

\int_4^6 (0.1x+k)\,dx + \int_6^8 0.3\,dx = 1

. The first integral is

0.05(6^2-4^2) + 2k = 1 + 2k

and the second is

0.6

, so

1 + 2k + 0.6 = 1

and

k = -0.3

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

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