GATE DA 2026 Set 1 — Question 53
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Probability & Statistics → Continuous Random Variables → Chi-Squared Distribution
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Question
Let be independent random variables. Each of the random variables follows distribution. Define .Which of the following statements is/are correct?
Correct answer
(A) Σᵢ₌₁ⁿ Xᵢ² follows Chi-square distribution with n degrees of freedom.; (B) Σᵢ₌₁ⁿ (Xᵢ - X)² follows Chi-square distribution with (n-1) degrees of freedom.; (C) X₁² + Xₙ² follows exponential distribution with mean 2.
Solution
Given are i.i.d.Option (A): The sum of squares of independent standard normal variables follows a Chi-square distribution with degrees of freedom. . Correct.Option (B): This is a standard result for sample variance of normal populations. . Correct.Option (C): and . Since they are independent, their sum . The PDF of is . For , , which is the PDF of an Exponential distribution with rate . The mean is . Correct.Option (D): . Thus . The square follows a Chi-square distribution with 1 degree of freedom, not 2. Incorrect.
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