And now I'm going to be really demanding.

What I want to do in this video is to generalize it.

That's largely influenced by the sample size.

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

At 10 they also just both multiply.

Compute for me the mean and the variance using the maximum likelihood estimator I just gave you.

In this case it's the same, and in this case it's 0.

I'm giving you a skeleton program, which has a function update, that takes as an input a mean and a variance for the first distribution and a mean and a variance for the second distribution and outputs the new mean and the new variance of the product of those.

In the 1 dimensional case, it is mu and sigma squared.

These were the results for Jeremy Lin.

The variance is a measure of how far the data is spread.

Mean of the distribution

And yes, it's as easy as this. We just add the two means and the two variances.

The prior has a mean of mu and a variance of sigma squared.

I gave you the formulas for how to compute the mean of the outcome.

Let me just briefly summarize what we're doing.

If the probability distribution of x is Gaussian, it would mean Mu and variant sigma squared, then we'll write this as x the random variable tilde.

I've got in my example minus the mean, minus the center, minus the mean of distribution.

Let's practice this. I'll give you a couple of quizzes now.

These are often called multivariate gaussians.

Linear middle of the distribution

There are 4 possibilities 0 and 0, 1 and 0, 0 and 1, and 1 and 1.

Suppose I use the following Gaussians μ 10, σ 2 4, ν 12, r 2 4

Here is an interesting data sequence 3, 3, 3, 3, 3.

In particular, consider a single data item as our sequence.

Since on average it didn't move, we just add the mean of its starting position to the no movement, and we get a mean of 0.

Mean value of the normal distribution

I finally want to teach you about the concept of a standard score.

They'll fluctuate, right.

And so these are going to be three values drawn from a Gaussian distribution with mean zero and variance or standard deviation equal to one.

MV is the linear middle of the distribution.

It's one standard deviation in the distribution of sample means. So this distribution of sample means, again.

This is what you do, for example, when you compute the mean outcome of an election from a sample of size N. You take the sum of the things you saw and you compute the mean.

And again, start to embrace this notation, this is gonna be very ah, common, and we'll use it a lot as we move forward through the course.

Average

Otherwise it doesn't work.

Let's now go back and write a program in which we calculate the new mean and the new variance term.

Our Xi's can now be arbitrary values not just 0s or 1s.

Then we can sum the squared deviations scores and divide by n. So if we sum this column we get 922.1 divided by ten is 92.21.

Remember the notation, the sum of squares and mean squares.

So the T statistic that we could derive from that is going to be our mean, 17.17 minus the true mean of our population.

Suppose we multiply two Gaussians as in Bayes rule a prior and a measurement probability.

Mean value of the standard logarithmic distribution

The three main principles are what we just we worked through.

I get a parallel data set called ndata that is data minus the mean, squared.