impossiblewizardry | (no subject)

OK let's say you've got an event:

X > Y

That is, one random variable is higher than another. This is equivalent to

V > c

where V is a random variable and c is a constant.

V has the same mean as X

c is the mean of Y

The variance of V is the variance of X plus the variance of Y

Oh, this also requires that X and Y are independent.

ANyway... basically we're moving all the "randomness" to the left hand side, and we end up comparing a random variable to a constant cutoff, rather than two random variables to each other

There's lots of ways to do that (for example X-Y>0) but I like this one cause visually, the densitives of X and V are in the same place, and if I imagine increasing the mean of X, they both shift to the right. And I can imagine a mark on the x axis at the mean of Y