Sigma Multiplied Gaussian Distribution

[jaydnul]

Hi!

Say i have two variables that have independent gaussian distributions of probability of being a certain value when i sample them, what is the likely hood that both will land on a 3 sigma value simultaneously? Is there an equation that easily determines that? Also for other combinations like one at 2 sigma and the other at 3 sigma, or what if i have 3 variables instead of two, etc.

Side question, if i had applied a sigma multiplier of 2x, how does that affect the values?

Thanks!

 

[mathman]

Prob. of a specific value is 0. Rephrase question - use ranges.

 

[FactChecker]

Hi!

Say i have two variables that have independent gaussian distributions of probability of being a certain value when i sample them, what is the likely hood that both will land on a 3 sigma value simultaneously?

I will assume that you mean within 3 sigma from the mean.
The probability of a result being within 3 sigma is 0.9974. The probability of two independent, identically distributed results being within 3 sigma is 0.9974 * 0.9974 = 0.9948

Is there an equation that easily determines that?

The equation for a result being within a certain sigma from the mean is complicated but there are online calculators and tables that you can use. Here is one. It gives you the probability of a result being between 0 and Z, so you would want to double that to include results between -Z and 0. Once you have a probability, $p$, from the online link and want to use it for $n$ Independent, Identically Distributed (IID) results, the formula is $(2 p)^n$.

Also for other combinations like one at 2 sigma and the other at 3 sigma, or what if i have 3 variables instead of two, etc.

The probability of a result being within 2 sigma is 0.9544. The probability of two independent, identically distributed results being within 2 sigma is 0.9544 * 0.9544 = 0.9109

The probability of three results within 3 sigma is 0.9974 * 0.9974 * 0.9974 = 0.9922
The probability of three results within 2 sigma is 0.9544* 0.9544* 0.9544= 0.8693

Side question, if i had applied a sigma multiplier of 2x, how does that affect the values?

It's not clear to me what you mean by a "sigma multiplier". The formula for any number of sigma values is complicated.