Question

If \( X \) is a continuous random variable whose probability density function is given by \[ f(x) = \begin{cases} k(4x - 2x^2), & 0<x<2 \\ 0, & \text{otherwise} \end{cases} \quad \text{then the value of } k \text{ is:} \]

• A. \( \frac{3}{8} \)
• B. \( \frac{5}{8} \)
• C. \( \frac{7}{8} \)
• D. \( 1 \)

Hint:

To determine a constant in a PDF, always integrate over the support and set the integral equal to 1.

Answer 1

The Correct Option is A

To find \( k \), use the fact that the total area under a probability density function must equal 1: \[ \int_0^2 k(4x - 2x^2) dx = 1 \] \[ k \int_0^2 (4x - 2x^2) dx = k \left[ 2x^2 - \frac{2}{3}x^3 \right]_0^2 = k \left( 8 - \frac{16}{3} \right) = k \cdot \frac{8}{3} \] \[ \Rightarrow \frac{8k}{3} = 1 \Rightarrow k = \frac{3}{8} \]