Normal Distributions

Let's contruct the formula for the bell curve function. We're working with the mean and variance.

The graph of this first forumla is:

As expected : when x=mu, then f(x) = 0. Also it's quadratic to it has a parabolic shape.

If we devide by sigma^2, this affects how wide or narrow it is:

If sigma = 4, then our curve will become wider. The larger our variance is, the larger the quadratic. This is because for a large sigma, the resulting f(x) will be smaller. So sigma will affect our resulting bell curve. This must be taken into the formula so that's why we added it.

Say we multiply by -1/2, what happens is that the quadratic is flipped over, so it becomes the green curve.

We need this because our bell curve is... bell curve. We need to multiply be a neg number to get it, otherwise it's the wrong shape. But this isn't enough to have our bell curve everything is under 0.

Now, if we add exponent: Where is f(x) maximized?

Answer = mu. This function is optimized when whatever is in the power is the largest. And that happens when mu = 0, so when x=mu.

Also, how can we find the maximum? Because it's when when x= mu, then f(mu) = e^0 = 1. The max height of the curve is 1.

This function is also minimized when x = + or - infinity.

When x = infinity of - infinity, f(x) = 0... which is exactly what we want our bell curve to be!

So the normal distribution function should be:

However, the area under the bell curve with this formula is not one. That is problematic because we want it to be one. The area is actually sqrt(2piesigma^2).

Therefore, in order for the bell curve's area to be equal to one, we multiply the formula by a normalizer.

This means the the bell curve function is described by:

In summary

To interpret this, we find inside the exp the quadratic penalty term of deviations from the expectation of the mean of the expression, and then the exponential squeezes it back into the bell shaped curve.

So the probability of X p(X) is twice as likely as the P(X') if the height of X on the curve is twice the heigh of X'.

This means that the height of the curve is proportional to the probability of this value being drawn.

Depending on the amount of coin flips we have, we use one or the other formulas. The last one is called our Gaussian exponential. 2nd one is binomial distribution.

The purpose of normal distribution is to be able to tackle many coin flips, or whatever example. So when we do hypothesis testing, and intervals, etc, we'll do it for the Gaussian exponential and not for binomial distributions or coin flips.