A continuous random variable $X$ has probability density function $f(x) = k x (2 - x)$ for $0 \le x \le 2$, and $f(x) = 0$ elsewhere. The value of the constant $k$ is ________ (round off to 2 decimal places).
Answer Key:
0.75
Step-by-Step Solution & Explanation
By the normalization property of probability density functions:
$\int_{-\infty}^\infty f(x) \, dx = 1$
$\int_0^2 k(2x - x^2) \, dx = 1$
Evaluating the definite integral:
$k \left[ x^2 - \frac{x^3}{3} \right]_0^2 = 1$
$k \left( 4 - \frac{8}{3} \right) = k \left( \frac{12 - 8}{3} \right) = k \left( \frac{4}{3} \right) = 1$
$k = \frac{3}{4} = 0.75$