Statistics Expected Value & Variance of Probability Distributions

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We are going to talk about expected value and variance of probability distribution.0001

Just a brief recap of discrete versus continuous random variable is0005

you need to understand random variable in order for us to move on to understanding expected value and variance.0016

Then we are going to do a brief mean and variance review to just think about all the different kinds of mean and variance we have learned so far.0023

We are going to talk about the new versions of mean and variance, mean and variance probability distribution.0033

We are going to talk about three special situations, linear transformations of the random variable x and what happens to mean and variance,0042

the sum of n independent values of x and the sum of difference of independent values of X and Y.0050

Drawn from two different random variable pools.0059

There are discrete versus continuous random variables so far we have been talking about random variable like x.0062

x could be sum of 2 die, x could be the sum of 2 TV show, x could just be something simple like number of people in a room.0073

It does not really matter what x is.0118

X is whatever random variable and then whatever that random variable is you have the probability of those x in your probability distribution.0120

That is what we have been looking at are discrete random variable.0130

These are called discrete, their numbers like when we have the sum of two die, their numbers like 2, 120137

but they are discrete they are not continuous because having 1.7 as an expected value that means that if we had a distribution that look at 1, 2, 3, 4 these are our x.0147

Having 1.7 although there is meaning it is like on average like somewhere around here for our distribution you are getting 1.7.0170

Is it not possible to get that 1.7.0178

Let us say we have the sum of two die, let us say that ended up an expected value of uneven set of two die and end up being 4.7.0184

That is a perfectly fine expected value but can you ever roll an actual sum of 4.7,0197

no that is impossible and because of that those are called discrete random variable.0206

There are bins that you have to follow and those are the only possible values for that random variable.0212

Now random variables like height are continuous because in something like average height or the sum of heights.0222

It is not that I have particular values that you can have.0232

You could have all kinds of different values.0236

It will be less likely than others but there are infinite number of possibilities in between 2 discrete sums.0239

We are only trying to be talking about discrete random variable and the probability distributions.0248

So far of discrete random variables.0257

All the stuff we have learned about expected value and all that stuff, it only works for discrete random variable.0267

Later on will learn about how to deal with continuous random variables and that can be exciting if it can open up all the world for us.0274

Given that, now we could do a brief review of mean and variance and we talk about samples, population and we are adding on probability distribution.0283

With samples remember the mean is going to be symbolized by X bar while in a population the mean is symbolized by mu.0296

Here the mean is symbolized by the expected value of x or mu sub x.0304

First off, you know that the symbols are different, but the sample and population what you end up having to do is summing all the x.0313

I=1 to n then dividing by how many number you have.0324

In populations, what you end up doing is basically the same thing except you just change the notation slightly0339

so that it reflects that you are doing this for the entire population, not just your little subset sample.0348

Here we have x sub i just like before but instead i going from 1 to n, we have from 1 to N and divide by N and the number of all different people in your population.0357

At first glance you might think what we learned about expected value might look a little bit different to you0377

because you have the sum of all these different values of all these different x times the probability of those x.0385

you might think we are not like adding them up and dividing by n, but in fact we are because let us unpack the p(x).0396

If you think about the probability of X, think about double-clicking on it and we open it up what is actually inside?0405

What is actually inside is the number of x, the number of times where you will get x out the total frequency.0415

We are looking at something divided by the total frequency of however many it is but we are also weighting it by the number of x.0444

Before we had things like 1 out of 36, there are 36 possible outcome and the number of times where you will get 1-1 is just 1out of 36.0456

We can change that into a probability or we could unpack it as the number of times you will get x out of the total frequency, the total number of outcomes.0470

It will be a little bit more transparent.0490

In that way we have in your weighting x by however frequent it is and dividing by the total.0493

That is very similar to our notion of mean like all these x divide by some total number of something.0503

In that way we have that idea still present here we just have to unpack it a little.0513

Let us talk about variance.0518

Before what we want from variance was something like average distance away from the mean.0521

We want these points and we want to know through the average distance away from the mean and we could not just look at deviations away from the mean0530

because when you have x - X bar sometimes we positive and sometimes the only negative, so that is adding up to 0.0539

We will square everything right.0548

Here and in sample the variance we called s² actually.0553

Here we would call this Sigma² and let us start here.0562

The Sigma² what we are going to do is just take all of the difference squared deviations away from the mean.0568

Take all these x and get all their squared deviations away from the mean, and then divide by N, how many x we have.0580

It is the same i from 1 to N.0595

When we looked at variance and more importantly standard deviation, which is going to be the square root of the things,0598

what we wanted was the same idea, so squared deviation and this time we use x bar².0610

Now we need to do a little bit of a correction and so we divide by n-1.0620

In order to get standard deviation we just square root both sides and we get as square roots of x sub i – x bar² over n -1.0629

You could also square that square root both sides here and we get Sigma equals the square root of sum of squared deviation away from mu this time over n.0649

Let us sample population, but now let us talk about in probability distribution.0669

Just like how here you see how mu is like population because probability distribution theoretical, so we use those Greek letters.0678

Here we will use sigma, square that for variance but we will also put that x there.0694

Instead of expected value we call this variance.0702

You could write it as bar x or sigma sub x².0705

If you break this down you could see the similarity but once again I will put it in the probability form where now you are summing the squared deviation.0714

It is x - and imagine what you would pick here, you would not put mu or X bar, you would put it corresponding mean, which is mu sub x² times the P(x).0724

You multiply all these together.0755

It is the same thing if you break it apart you could see sort of this piece and this piece.0757

This piece is very similar here and we are using probability to weight each x and then divided by the total number of outcomes.0764

You could see this part is very similar to these parts and once again you could break down p(x)0775

to be in a number of outcomes that look like X over the total number of outcomes.0786

You could break it down, but here I'm going to write the standard deviation form.0794