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

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

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### Video Transcript

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

Weβve been given some information for two independent random variables π₯ and π¦. We know the variance of π₯ is 24 and the variance of π¦ is equal to 30. Weβre looking to determine the variance of seven π₯ plus nine π¦. So we begin by recalling that for two independent random variables π΄ and π΅, the variance of their sum is the sum of their variances. So the variance of π΄ plus π΅ is the variance of π΄ plus the variance of π΅.

But we also know that the variance of some multiple of a random variable π₯ β letβs call that π of π₯ β is equal to π squared times the variance of π₯. So what weβre going to do is split our variance, the variance of seven π₯ plus nine π¦, up into the sum of the variances, so variance of seven π₯ plus the variance of nine π¦.

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