sigma

sigma

Sigma

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Gaussians.

So those would be the high peaks of the density field.

We're Sigma Zeta Chi.

So what is the distribution of all data points and just those that are higher than 1 sigma above the mean?

So if you take this data set and if you estimate mu1, mu2, sigma1, sigma2, this is what you get.

So, just be careful whether you are using sigma 1, sigma 2, or sigma squared 1 or sigma squared 2.

And that notation is Sigma notation.

And that just gives me the center of this distribution. How about sigma squared?

The Sigma Tau Omega team and the Acacia team.

One lap to go!

You'll see structures that are corresponding to higher contrast as you go to deeper redshifts.

Still leading, Sigma Tau Omega, team number 1.

X1 through X M, we then fit the parameters, mu 1 through mu n, and sigma squared 1 through sigma squared n, and so these were the formulas similar to the formulas we have in the previous video, that we're going to use the estimate each of these parameters, and just to give some interpretation, mu J, that's my average value of the j feature.

It's not.

Again, there's variance in the population, sigma squared.

And sometimes it's easier to think in terms of sigma.

Speaking of flowers, I'm putting together a party after the party at Kappa Eta Sigma.

Here I'm using sigma because I'm talking about population parameters.

I'm think about standard deviations, but the formula on the previous 5 actually gave the estimates of the squares of theses things, so sigma squared 1 and sigma squared 2.

Now, why is this neat, or why is this cool at all?

Well, first of all, we found out a way to sum this

Cos just like you, they wanna be Sigma Zeta Chi.

And so, maybe this is a reasonable estimate of

Gaussian density is this, 1 over 2pi, sigma e to the negative x minus Mu squared over 2 sigma squared.

And the width of this Gaussian is Gaussian thus controlled by sigma by that variance parameter sigma.

And in pictures what p of x1 parametrized by mu1 and sigma squared 1 and p of x2, parametrized by mu 2 and sigma squared 2, that would look like these two distributions over here.

So the standard deviation is one half and the variance sigma squared would therefore be the square of 0.5 would be 0.25.

As a result of that terrible exchange, the Cutter team has dropped from the lead.

So p of x, that is the height of this plot, is

So this curve here, this is really p of x plotted as a function of x, you know, for a fixed value of Mu and of sigma squared sigma squared, that's called the variance.

left, that is what you get if you take a fixed value of Mu and a fixed value of sigma and you plot p of x.

Let's say Mu is equal to zero and sigma is equal to one half.

We can multiply by that term once rather than having it in each row of the table.

So it's just e to the, this whole thing over here, minus x minus the mean squared over 2 sigma squared.

literally just p of x1 parametrized by mu 1 sigma squared 1, times p of x2 parametrized by mu 2 sigma squared 2.

Well, the other type of bond, so this right here let me make this clear.

This sp3 orbital is pointing that direction, and they're overlapping right around here.

And so, this bell shaped curve is parameterized by those 2 parameters Mu and sigma.

I've never seen it written this way but it gives me a little intuition that sigma squared, it's always written as sigma squared, but it's really just the variance and the variance is what you calculate before you calculate the standard deviation, so that's interesting.

By doing this, we're doing less work.

That's the small lobe, and then that's the big lobe like that.