This is a bit of a nit for me, but the variance of a sum is not generally the sum of variances! The equation in this slide only holds when each Y_i is independent from each other. In general, the variance of a sum is the sum of covariances, which may involve more terms than just the variance of each of the random variables being summed. I'm assuming this is not that important for the context of this class given that we're given this formula on the slide, but I just wanted to make a note since it's can be easy to forget :')
saif-m17
Another point I will add on top of @brandonlouie's point is that the formula at the bottom only holds if the Y_i are independent as well (as they explained the variance of a sum of random variables is not equivalent to the sum of the variances of those random variables unless they are independent). In the context of monte carlo integration for this class, I guess since we assume the x_i we draw are iid, both of these facts are reasonable to assume. Also, another property of Variance is that the Var(X + c), where c is a constant and X is a random variable is just equivalent to Var(X). Since variance is a measure of spread, shifting the distribution doesn't change the spread of it.
This is a bit of a nit for me, but the variance of a sum is not generally the sum of variances! The equation in this slide only holds when each Y_i is independent from each other. In general, the variance of a sum is the sum of covariances, which may involve more terms than just the variance of each of the random variables being summed. I'm assuming this is not that important for the context of this class given that we're given this formula on the slide, but I just wanted to make a note since it's can be easy to forget :')
Another point I will add on top of @brandonlouie's point is that the formula at the bottom only holds if the Y_i are independent as well (as they explained the variance of a sum of random variables is not equivalent to the sum of the variances of those random variables unless they are independent). In the context of monte carlo integration for this class, I guess since we assume the x_i we draw are iid, both of these facts are reasonable to assume. Also, another property of Variance is that the Var(X + c), where c is a constant and X is a random variable is just equivalent to Var(X). Since variance is a measure of spread, shifting the distribution doesn't change the spread of it.