For independent random variables X, Y, what is Var(X+Y)?

$\sigma_X^2+\sigma_Y^2+2\sigma_{XY}$

$\sigma_X^2+\sigma_Y^2+\sigma_{XY}$

$\sigma_X^2+\sigma_Y^2-2\sigma_{XY}$

GATE 2021 MN – Question 24

Engineering Mathematics · Probability and Statistics · 1 mark · Multiple choice

For independent random variables X, Y, what is $\operatorname{Var}(X+Y)$?

$\sigma_X^2+\sigma_Y^2$
$\sigma_X^2+\sigma_Y^2+2\sigma_{XY}$
$\sigma_X^2+\sigma_Y^2+\sigma_{XY}$
$\sigma_X^2+\sigma_Y^2-2\sigma_{XY}$

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Correct answer: (A) $\sigma_X^2+\sigma_Y^2$

Explanation

Generally, $\operatorname{Var}(X+Y)=\sigma_X^2+\sigma_Y^2+2\operatorname{Cov}(X,Y)$. Independence makes covariance zero, leaving the sum of variances, A. Standard deviations themselves do not generally add.

GATE 2021 MN – Question 24

Explanation

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