GATE 2024 ST – Question 51

Joint Distributions · Joint and Conditional Distributions · 2 marks · Multiple select

(X,Y) has joint density 3/4 on {−1≤x≤1,x²≤y<1},zero elsewhere. Which hold?

X has the same distribution as −X.
E(Y|X=0)=1/2
Corr(X,Y)=0
X,Y are independent.

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Show answer and explanation

Correct answer: (A) X has the same distribution as −X.; (B) E(Y|X=0)=1/2; (C) Corr(X,Y)=0

Explanation

The support and density are symmetric in x, giving A. Conditional on X=x, Y is uniform(x²,1), so at x=0 its mean is 1/2. Symmetry makes E X=0 and E(XY)=0, hence zero covariance and correlation. But the conditional range and mean (1+x²)/2 depend on x, so they are not independent. **A,B,C**.

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