Thread: Comparison of two normal curves

Hello, I was curious how to find the probability that a random number generated in one normal curve will be greater than a random number generated in a second normal curve.

For example, if you have a curve with m=1, st dev = 0.5 and a second curve with m=2, st dev = 1, what is the probability that a random number from the first curve will be greater than a random number from the second curve?

Thank you for your help.

Hmmm. I would let Z1 and Z2 be independent random numbers from two different normal curves.

Then let Y=Z1-Z2. Y is also normal(mean1-mean2,var1+var2), and you could check P(Y>0) to see if Z1>Z2.

In your example, Z1 is N(1,.25) and Z2 is N(2,1). Then Y is N(-1, 1.25) (remember to add variance not sd). If you standardize this, you get P(Y>0)=>P(Z>.894) = about .186 which makes sense as Z2 is generally going to be larger.

I also believe that if your means are the same, it will always be a 50/50 chance for Z1>Z2 regardless of variance due to symmetry; both Z1 and Z2 have equal chances of being above or below their common mean.

Thanks, good thinking. That makes a lot of sense.