2 marksStatisticsRandom variables and standard distributionsUniform distributionsModerateabout 2 min

GATE 2023 Question 62: Random variables and standard distributions

Let $X$ be a random variable with the probability density function $f(x)$ such that

$f(x) = \begin{cases} \dfrac{1}{2\sqrt3} & \text{if } -\sqrt3 < x < \sqrt3 \\ 0 & \text{otherwise} \end{cases}$

Then, the variance of $X$ is ______ (in integer).

Answer and worked solution

Answer: 1

This density describes a Uniform distribution on the interval $(-\sqrt3, \sqrt3)$ : it's constant ( $\frac{1}{2\sqrt3}$ ) over the interval and zero elsewhere. Checking normalization: the interval width is $\sqrt3-(-\sqrt3)=2\sqrt3$ , and $\frac{1}{2\sqrt3}\times 2\sqrt3=1$ , confirming it's a valid density.

For a Uniform $(a,b)$ distribution, the variance formula is:

$Var(X) = \frac{(b-a)^2}{12}$

Here $a=-\sqrt3$ , $b=\sqrt3$ , so $b-a = 2\sqrt3$ , and $(b-a)^2 = (2\sqrt3)^2 = 4\times 3 = 12$ .

$Var(X) = \frac{12}{12} = 1$

Therefore the variance of $X$ is 1.

The trap: 3

Students sometimes recall the variance formula for a symmetric interval around zero, $(-a,a)$ , as $a^2/3$ rather than the general Uniform $(a,b)$ formula $(b-a)^2/12$ — applying $(\sqrt3)^2/3=1$ actually coincides here by a specific-case shortcut, but a student who instead misremembers the shortcut (e.g. using $a^2$ directly without dividing) could land on 3 instead of 1.

Plain-text version

Let X be a random variable with the probability density function f(x) such that f(x) = { 1/2√3 if -√3 < x < √3; 0 otherwise } Then, the variance of X is (in integer).

Also written as: Let X be a random variable with the probability density function f(x) such that f(x) = { 1/2sqrt(3) if -sqrt(3) < x < sqrt(3); 0 otherwise } Then, the variance of X is (in integer).

Random variables and standard distributions has come up 173 times in 7 exams since 2005.

What this question tests

This question is on Uniform distributions, part of Random variables and standard distributions in Statistics.

Equally likely outcomes over a finite set or an interval.

TestStop has 14 past-paper questions on this subtopic.

Take this paper as a timed mock More on Uniform distributions

Question from the GATE 2023 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.