Blake O'Hare
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Author: Ushanka

Integral for Standard Normal Distribution Curve

This is the equation for a normal distribution curve, one of the more important functions in probability and statistics:

[equation]

It is the very recognizable bell-curve shape.

[equation]

In the equation above, μ is the expected value of the distribution. It also corresponds to the point where the function has the greatest y-value. σ is the value of one standard deviation. Ask...ask your stats instructor. When μ=0 and σ=1, we get what is called the standard normal distribution curve:

[equation]

The area under this standard normal distribution curve is equal to exactly 1. I am going to show you how to integrate this, because I think it is the coolest integral I have ever seen.

Our limits of integration are plus and minus infinity. We'll say the area under the SNDC is A.

[equation]

Oh, geez, that is a mean ol' integral. That would be a real pain to solve. Sure, I guess Wolfram Alpha can do it, but that is stupid and lame and stupidly lame. We're going to do this the cool person's way. You do want to be a cool person, right? Right. Now, let's note something obvious: the SNDC in the x-y plane will have the same area A as the SNDC in the x-z plane, but the SNDC will have the equations

[equation]

instead. Similarly, the SNDC in the x-z plane will have the same area A as the SNDC in the y-z plane, but it will have a different variable of integration.

[equation]

Now, since we've got two integrals that both equal A, we can multiply them together to get a double integral that equals A2.

[equation]
[equation]

This gives us a three-dimensional bell curve, a surface in space that looks like this:

[equation]

(Apologies for the two missing latitude/longitude lines. I'm not exactly sure what happened to them, but I suspect they escaped my picture and fled to Venezuela.) We can observe from the visual representation of the surface that this is a thing which could be represented in polar coordinates. Here's a quick refresher of Important Polar Coordinate Facts.

[equation]

I guess the second of those Important Facts is unnecessary for this integral, but it is good to know nonetheless. So, when we make the appropriate substitutions, we get:

[equation]

Since there's no θ in the formula itself, that one is pretty easy to integrate.

[equation]

Now this is a nice integral. We can use substitution and quickly get the following:

[equation]

I assume you're good enough with the limits and the second part of the Fundamental Theorem of Calculus to follow my logic in this next step.

[equation]

I assume you're good enough with arithmetic and spatial reasoning to follow my logic in this next step.

[equation]

SOME COOL STUFF

If you try this again with a (nonstandard) normal distribution curve, you will see that the area is still 1. Yep, try it and see!

Now that you've finished reading this post, what are you going do? You should go join the Forum.
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