Question Bank • QB_EM_112 NAT • 2 Marks

Section 1: Engineering Mathematics

Probability and Statistics

Subtopic: Random variables

Question Statement:

A continuous random variable $X$ has probability density function $f(x) = k x (2 - x)$ for $0 \le x \le 2$ and $0$ elsewhere. Calculate the value of the constant $k$.

Answer Key:

0.75

Step-by-Step Solution & Explanation

For a valid probability density function, $\int_{-\infty}^\infty f(x) dx = 1$: $k \int_0^2 (2x - x^2) dx = 1$ $k \left[ x^2 - \frac{x^3}{3} \right]_0^2 = 1$ $k \left[ 4 - \frac{8}{3} \right] = k \left( \frac{4}{3} \right) = 1$ $k = \frac{3}{4} = 0.75$
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