The Standard Normal Distribution Fr. Chris A.P.Statistics St. Francis High School

Requirements for Any Probability Distribution Always Positive Total area under the curve must be 1 (Since 1 represents 100%)

The Standard Normal Distribution Any value of x will produce a POSITIVE result

But is the area below equal 1?

In other words, does YES!

Lets work it out…

But there is a problem… Has NO antiderivative! But Karl Friedrich Gauss figured a tricky way around this!

Let us introduce an “I” such that: I = So the area under the “Standard Normal Curve” would be

But let’s concentrate on the I I = Since x is merely a variable of integration, we can also express I as I =

So why not express I 2 as or

This double integral is actually the volume under a 3-D “bell”

Rectangles aren’t the only thing that integrates! We can now do a clever change of variable by converting to Polar coordinates!

How polar coordinates work, and how to make the switch... Any integral can be computed by the limit of Riemann sums over Cartesian rectangles or Riemann sums over polar rectangles. The area of a Cartesian rectangle of sides dx and dy is dx*dy but the area of a polar rectangle of sides dr and dt is NOT just dr*dt

Rather dA is a bit more... Getting back to our story...

Recall the Pythagorean Theorem? We can now do a clever change of variable by converting to Polar coordinates!

So now we DO have an antiderivative!

And because of symmetry We can now start evaluating the integral from negative to positive infinity…

Almost there!

But that wasn’t “I” So...that was “I” squared!

Recall the area under the Standard Normal Curve So…

So the area under the curve is 1! Wasn’t Professor Gauss clever? It is no accident that many Mathematicians still prefer to call the “Standard Normal” distribution The Gaussian Distribution