# Bounds of Q(x)

iVenky
I think everyone knows that

Q(x)= P(X>x) where X is a Gaussian Random variable.

Now I was reading about it and it says that Q(x) is bounded as follows

Q(x)≤ (1/2)(e-x2/2) for x≥0

and

Q(x)< [1/(√(2∏)x)](e-x2/2) for x≥0

and the lower bound is

Q(x)> [1/(√(2∏)x)](1-1/x2) e-x2/2 for x≥0

Can you tell me how you get this?

Thanks a lot.

Last edited:

One example for the first inequality: It is exact at x=0, as you can check. For $0<x<\frac{1}{\sqrt{2pi}}$, the derivative of the upper estimate is larger (negative with a smaller magnitude) than the derivative of Q(x), which is simply the normal distribution. Therefore, the upper estimate is valid.