1 markStatisticsJoint distributions, covariance and correlationVariance of sums and portfolio applicationsModerateabout 1 min

IIT JAM 2021 Question 47: Joint distributions, covariance and correlation

Let $X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$ with $\mu_1 = 2$ , $\mu_2 = 3$ , $\sigma_1^2 = 4$ , $\sigma_2^2 = 9$ and correlation coefficient $0.5$ . The variance of $(X_1 + X_2)$ is _______ (in integer).

Answer and worked solution

Answer: 19

The variance of a sum is $\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2$ .

Here $4 + 9 + 2(0.5)(2)(3) = 13 + 6 = 19$ . The answer is $19$ .

Plain-text version

Let X₁ ~ N(μ₁, σ₁²) and X₂ ~ N(μ₂, σ₂²) with μ₁ = 2, μ₂ = 3, σ₁² = 4, σ₂² = 9 and correlation coefficient 0.5. The variance of (X₁ + X₂) is _ (in integer).

Also written as: Let X_1 ~ N(mu_1, sigma_1^2) and X_2 ~ N(mu_2, sigma_2^2) with mu_1 = 2, mu_2 = 3, sigma_1^2 = 4, sigma_2^2 = 9 and correlation coefficient 0.5. The variance of (X_1 + X_2) is _ (in integer).

Joint distributions, covariance and correlation has come up 61 times in 7 exams since 2005.

What this question tests

This question is on Variance of sums and portfolio applications, part of Joint distributions, covariance and correlation in Statistics.

The variance of combinations of variables and where it is used.

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