Let X be a discrete random variable that is uniformly distributed over the set \-10,-9,,0,,9,10\. Which of the following random variables is/are uniformly distributed?

GATE 2024 EE – Question 29

Engineering Mathematics · Probability and Statistics: Random variables · 1 mark · Multiple select

Let $X$ be a discrete random variable that is uniformly distributed over the set $\{-10,-9,\cdots,0,\cdots,9,10\}$. Which of the following random variables is/are uniformly distributed?

$X^2$
$X^3$
$(X-5)^2$
$(X+10)^2$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $X^3$; (D) $(X+10)^2$

Explanation

A one-to-one function of a uniform discrete variable is uniform. $X^3$ is one-to-one, so uniform. $(X+10)^2$: $X+10$ takes the values $0,\dots,20$, all non-negative, so squaring is one-to-one: uniform. $X^2$ takes the value 0 with probability $1/21$ and every other value with probability $2/21$, and $(X-5)^2$ takes values $\{0,\dots,5\}$ twice and larger values once: not uniform.