Integrating the normal probability density function

[ch1no]

Hello,

I'm wondering if any of you can help explain a part of the solution to my problem. The original problem is as follows:

> For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.

The problem is easily solved by looking at probability tables. That's not the point here. My question concerns changing limits of integration:

How did the limits of integration on top turn into the ones on the bottom? I've tried algebraically manipulating the terms. I haven't been able to discover the logic yet.........

 

[royhaas]

It is algebra. Substitute \(\displaystyle x = \mu + 2\sigma\) in the expression \(\displaystyle t = (x-\mu)/\sigma\).