The GATE Grind

GATE 2025 CH – Question 37

Engineering Mathematics · Probability and Statistics: Random variables, Poisson, Normal and Binomial distributions · 2 marks · Multiple choice

A probability distribution function is given as

$p(x) = \begin{cases} \frac{1}{a}, & x \in (0, a) \\ 0, & \text{otherwise} \end{cases}$

where $a$ is a positive constant. For a function $f(x) = x^2$, the expectation of $f(x)$ is

  1. $\frac{a^2}{3}$
  2. $\frac{a^3}{3}$
  3. $\frac{2a^2}{3}$
  4. $\frac{2a^3}{3}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\frac{a^2}{3}$

Explanation

$E[x^2] = \int_0^a x^2\frac{1}{a}dx = \frac{1}{a} \cdot \frac{a^3}{3} = \frac{a^2}{3}$.