GATE 2026 EC Q.38: solution and answer

Official answer: ASource: Official answer key, GATE 2026 EC (IIT Guwahati)

1. Here "correlation" means the correlation $R_{XY}=E[XY]$, as used in signals and communication. It is not the normalised correlation coefficient. (Step 6 checks this reading.)

2. Moments of $X\sim U(-1,1)$: the density is $\frac{1}{2}$ on $[-1,1]$. The density is symmetric about zero, so every odd moment is zero: $E[X]=0$ and $E[X^3]=0$. Also $$E[X^2]={\int }_{-1}^1\frac{x^2}{2}dx=\frac{1}{2}\cdot \frac{2}{3}=\frac{1}{3}$$

3. Moments of $N\sim \mathcal{N}(0,1)$: $E[N]=0$. Because $X$ and $N$ are independent, $E[XN]=E[X]E[N]=0$.

4. Correlation of $X$ and $Y$, by linearity of expectation: $$E[XY]=E[X(X+N)]=E[X^2]+E[XN]=\frac{1}{3}+0=\frac{1}{3}$$

5. Correlation of $X$ and $Z$: $$E[XZ]=E[X(X^2+N)]=E[X^3]+E[XN]=0+0=0$$

6. Checking the reading: since $E[X]=0$, the covariances equal these values, $\text{Cov}(X,Y)=\frac{1}{3}$ and $\text{Cov}(X,Z)=0$. The normalised coefficient would be ${\rho }_{XY}=\frac{1/3}{\sqrt{1/3}\sqrt{1/3+1}}=\frac{1}{2}$, and $\frac{1}{2}$ is not in any option. So the question means $E[XY]$. Under either reading $X$ and $Z$ are uncorrelated, even though $Z$ depends on $X$ through $X^2$.

7. The pair is $\left(\frac{1}{3},0\right)$, which is option (A).