GATE 2024 MT – Question 11

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

Independent normally distributed variables $X_1,X_2$ have means $\mu_1,\mu_2$ and variances $\rho_1,\rho_2$. Their sum $X=X_1+X_2$ has mean $\mu$ and variance $\rho$ satisfying

$\mu=\mu_1+\mu_2,\ \rho=\rho_1+\rho_2$
$\mu^2=\mu_1^2+\mu_2^2,\ \rho=\rho_1+\rho_2$
$\mu=\mu_1+\mu_2,\ \rho^2=\rho_1^2+\rho_2^2$
$\mu^2=\mu_1^2+\mu_2^2,\ \rho^2=\rho_1^2+\rho_2^2$

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Correct answer: (A) $\mu=\mu_1+\mu_2,\ \rho=\rho_1+\rho_2$

Explanation

Expectation is linear, so μ=μ1+μ2.

Variance of a sum is ρ1+ρ2+2Cov(X1,X2); independence makes the covariance zero.

The sum is also normally distributed.

Therefore **A** gives both parameters correctly.