GATE 2026 DA Q.63: solution and answer

GATE 2026 · DAQ.63 · 2 marks · NAT

Let $X$ be a random variable that follows $Uniform (- 1, 1)$ distribution. The conditional distribution of the random variable $Y$ given $X = x$ is the $Uniform (x^{2} - 0.1, x^{2} + 0.1)$ distribution.

The value of $correlation (X, Y)$ is ______. (Answer in integer)

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The correlation of $X$ and $Y$ is $0$ .

Official answer: 0Source: Official answer key, GATE 2026 DA (IIT Guwahati)

Given $X = x$ , $Y$ is uniform on $(x^{2} - 0.1, x^{2} + 0.1)$ , an interval centred at $x^{2}$ . So $E [Y ∣ X = x] = x^{2}$ , that is $E [Y ∣ X] = X^{2}$ .
The correlation is $ρ = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}$ . Here $σ_{X} = 1/3 > 0$ . Also $σ_{Y} > 0$ , because $Y$ is not constant. So $ρ = 0$ exactly when $Cov (X, Y) = 0$ .
$X \sim Uniform (- 1, 1)$ has density $\frac{1}{2}$ on $(- 1, 1)$ , which is symmetric about 0. Hence $E [X] = \int_{- 1}^{1} \frac{1}{2} x d x = 0$ and $E [X^{3}] = \int_{- 1}^{1} \frac{1}{2} x^{3} d x = 0$ .
By the tower property, $E [X Y] = E [X E [Y ∣ X]] = E [X \cdot X^{2}] = E [X^{3}] = 0$ .
Then $Cov (X, Y) = E [X Y] - E [X] E [Y] = 0 - 0 \cdot E [Y] = 0$ .
Therefore $ρ = 0/ (σ_{X} σ_{Y}) = 0$ . This holds even though $Y$ depends strongly on $X$ through $X^{2}$ : the dependence is not linear, and correlation measures only linear association.

KEY IDEA

Find $E [Y ∣ X] = X^{2}$ , then use the tower property to get $E [X Y] = E [X^{3}] = 0$ . $X$ is symmetric about 0, so the covariance and hence the correlation are zero, although $Y$ depends on $X$ .

WATCH OUT FOR

Assuming that a strong dependence of $Y$ on $X$ (here through $X^{2}$ ) means a non-zero correlation. Correlation detects only linear association. Zero correlation does not imply independence.

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 4 Descriptive statistics questions, with notes

GATE 2026 DA Q.63: solution and answer

Answer and solution

More Descriptive statistics questions