Let X_1, X_2, , X_n be n independent random variables. Each of the random variables follows Normal( = 0, ^2 = 1) distribution. Define X = 1n_i=1^n X_i. Which of the following statements is/are correct?

_i=1^n X_i^2 follows Chi-square distribution with n degrees of freedom.

_i=1^n (X_i - X)^2 follows Chi-square distribution with (n - 1) degrees of freedom.

X_1^2 + X_n^2 follows exponential distribution with mean 2.

_i=1^n X_i^2 follows Chi-square distribution with n degrees of freedom.

_i=1^n (X_i - X)^2 follows Chi-square distribution with (n - 1) degrees of freedom.

X_1^2 + X_n^2 follows exponential distribution with mean 2.

(n\,X)^2 follows Chi-square distribution with 2 degrees of freedom.

GATE DA 2026 Q53: Chi-square facts for standard normal samples

Probability and Statistics
2 marks
MSQ
No negative marking

Let $X_1, X_2, \ldots, X_n$ be $n$ independent random variables. Each of the random variables follows $Normal(\mu = 0, \sigma^2 = 1)$ distribution. Define $\bar{X} = \dfrac{1}{n}\sum_{i=1}^{n} X_i$ .

Which of the following statements is/are correct?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, B, C (official GATE 2026 final answer key)

Concept used: A sum of $k$ squared independent standard normals is $\chi^2_k$ , and $\chi^2_2$ is the exponential distribution with mean 2.

First step: $\sqrt{n}\,\bar{X} \sim N(0, 1)$ , so its square is $\chi^2_1$ ; estimating the mean costs one degree of freedom in B.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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GATE DA 2026 Q53: Chi-square facts for standard normal samples

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