Question

A continuous random variable X that can assume values between x = 1 and x = 3 has a density
function given by f(x)=1/2.
(a) Show that the area under the curve is equal to 1.
(b) Find P(2 <X< 2.5).
(c) Find P(X ≤ 1.6).

Answer 1

(a) Area under the curve is

$= \int^3_1 f(x)dx = \int^3_1 \frac{1}{2} dx = [\frac{x}{2}]^3_1 \\ = \frac{3-1}{2} = \frac{2}{2} = 1$

(b) $P(2<X<2.5) = \int^{2.5}_2 \frac{1}{2} dx$

$= \frac{2.5-2}{2} = \frac{0.5}{2} = 0.25$

(c) $P(X≤1.6) = \int^{1.6}_1 f(x)dx = \int^{1.6}_1 \frac{1}{2} dx$

$= \frac{1}{2} (1.6 -1) = \frac{0.6}{2} = 0.3$