Question Bank • QB_EM_158 NAT • 2 Marks

Section 1: Engineering Mathematics

Probability and Statistics

Subtopic: Random variables

Question Statement:

A continuous random variable $X$ has probability density function $f(x) = 2x$ for $0 \le x \le 1$ and $f(x) = 0$ otherwise. Calculate the variance $\text{Var}(X)$ (round off to 3 decimal places).

Answer Key:

0.056

Step-by-Step Solution & Explanation

Expectation $E[X]$: $E[X] = \int_0^1 x (2x)\, dx = 2 \left[ \frac{x^3}{3} \right]_0^1 = \frac{2}{3}$ Second moment $E[X^2]$: $E[X^2] = \int_0^1 x^2 (2x)\, dx = 2 \left[ \frac{x^4}{4} \right]_0^1 = \frac{2}{4} = \frac{1}{2}$ Variance: $\text{Var}(X) = E[X^2] - (E[X])^2 = \frac{1}{2} - \left(\frac{2}{3}\right)^2 = \frac{1}{2} - \frac{4}{9} = \frac{9 - 8}{18} = \frac{1}{18} \approx 0.0556 \approx 0.056$
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