Video: Determining the Variance for the Sum of Two Independent Random Variables

Suppose π‘₯ and 𝑦 are independent, var(π‘₯) = 24, and var(𝑦) = 30. Determine var(7π‘₯ + 9𝑦).

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Video Transcript

Suppose π‘₯ and 𝑦 are independent, the variance of π‘₯ is 24, and the variance of 𝑦 is 30. Determine the variance of seven π‘₯ plus nine 𝑦.

We’ve been given some information for two independent random variables π‘₯ and 𝑦. We know the variance of π‘₯ is 24 and the variance of 𝑦 is equal to 30. We’re looking to determine the variance of seven π‘₯ plus nine 𝑦. So we begin by recalling that for two independent random variables 𝐴 and 𝐡, the variance of their sum is the sum of their variances. So the variance of 𝐴 plus 𝐡 is the variance of 𝐴 plus the variance of 𝐡.

But we also know that the variance of some multiple of a random variable π‘₯ β€” let’s call that π‘Ž of π‘₯ β€” is equal to π‘Ž squared times the variance of π‘₯. So what we’re going to do is split our variance, the variance of seven π‘₯ plus nine 𝑦, up into the sum of the variances, so variance of seven π‘₯ plus the variance of nine 𝑦.

Then, we know that the variance of seven π‘₯ is seven squared the variance of π‘₯. And the variance of nine 𝑦 is nine squared the variance of 𝑦. But we already saw that the variance of π‘₯ is 24 and the variance of 𝑦 is 30. So the variance of seven π‘₯ plus nine 𝑦 is seven squared times 24 plus nine squared times 30, which is 3,606. The variance of seven π‘₯ plus nine 𝑦 is 3,606.

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