Bounds of Q(x)

  • Thread starter iVenky
  • Start date
  • #1
iVenky
212
12
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:

Answers and Replies

  • #2
36,246
13,301
One example for the first inequality: It is exact at x=0, as you can check. For [itex]0<x<\frac{1}{\sqrt{2pi}}[/itex], 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.

In the same way, for all larger x, consider the limit of both for x->inf: It is 0. Now, the upper bound has a smaller derivative (negative with larger magnitude) everywhere, therefore it is valid there, too.

I would expect that you can get the other inequalities with similar methods.
 

Suggested for: Bounds of Q(x)

Replies
11
Views
587
Replies
2
Views
1K
Replies
17
Views
591
Replies
8
Views
444
Replies
3
Views
746
Replies
0
Views
747
Replies
16
Views
2K
Replies
15
Views
667
Top