Hi, X1 ~ N(mu, 35^2) and X2 ~ N(mu, 35^2) ---> X1 - X2 ~ N(mu - mu = 0, simga^2+sigma^2) Please could someone explain why the expectation of X1 - X2 is zero but the variance is 2 sigma^2? Thanks in advance
\(var(X - Y) = var(X) + var(-Y) = var(X) + (-1)^2 var(Y) = var(X) + var(Y)\) Intuitively, if you have two distributions with the same spread, you could have the largest value from the first distribution and subtract the smallest number from the second distribution - hence the range of results is twice as big. eg suppose \(X\) and \(Y\) are discrete distributions that take the values 3, 8 and 10 - which each have a range (spread) of \(10 - 3 = 7\). \(X - Y\) can take the values -7, -5, -2, 0, 2, 5 and 7, which has a range (spread) of \(7 - (-7) = 14\).
