A random variable \(X\) has a mean of 120 and a standard deviation of 15. A random variable \(Y\) has a mean
of 100 and a standard deviation of 9. If \(X\) and \(Y\) are independent, approximately what is the standard deviation
of \(X-Y\) ?
When adding or subtracting independent random variables, we need to sum up the sample variance rather than the standard deviation. Then, we can use the summed variance to obtain the standard deviation.
Recall that standard deviation is just the square root of variance.
$$ \sigma_{X-Y}^2 = \sigma_X^2+\sigma_Y^2 $$
$$ \sigma_{X-Y}^2 = 15^2+9^2 $$
$$ \sigma_{X-Y}^2 = 225+81 = 306 $$
$$ \sigma_{X-Y} = \sqrt{306} \approx \boxed{17.5} $$