Video Transcript
Suppose 𝑥 and 𝑦 are independent, the variance of 𝑥 is 24, and the variance of 𝑦 is 30. Determine the variance of seven 𝑥 plus nine 𝑦.
We’ve been given some information for two independent random variables 𝑥 and 𝑦. We know the variance of 𝑥 is 24 and the variance of 𝑦 is equal to 30. We’re looking to determine the variance of seven 𝑥 plus nine 𝑦. So we begin by recalling that for two independent random variables 𝐴 and 𝐵, the variance of their sum is the sum of their variances. So the variance of 𝐴 plus 𝐵 is the variance of 𝐴 plus the variance of 𝐵.
But we also know that the variance of some multiple of a random variable 𝑥 — let’s call that 𝑎 of 𝑥 — is equal to 𝑎 squared times the variance of 𝑥. So what we’re going to do is split our variance, the variance of seven 𝑥 plus nine 𝑦, up into the sum of the variances, so variance of seven 𝑥 plus the variance of nine 𝑦.
Then, we know that the variance of seven 𝑥 is seven squared the variance of 𝑥. And the variance of nine 𝑦 is nine squared the variance of 𝑦. But we already saw that the variance of 𝑥 is 24 and the variance of 𝑦 is 30. So the variance of seven 𝑥 plus nine 𝑦 is seven squared times 24 plus nine squared times 30, which is 3,606. The variance of seven 𝑥 plus nine 𝑦 is 3,606.
