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We have looked at using random variables individually in the past examples, so now we're going to look

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at the concept of using multiple random variables in a single experiment.

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So let it be a random variable with a PDF of our effects here.

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Also, let let's why be a random variable with a PDF of F..

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Why here it is now possible to define a PDF for the joint probability that is a probability of X and

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Y, and we define this as a probability density function of F, X, Y or for shorthand notation.

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We can just write it as F with X and Y being the parameters of the input.

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So we automatically know that this is a probability density function for random variables X and Y,

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so we can either use either of these annotations.

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So just like in basic probability, the joint probability gives us a probability of X and Y happening

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at the same time.

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So from the probability density function, the joint one, we can work out the joint probability and

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we can just do this by using our probability equation that we had before.

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But we're going to extend it to being a function of two variables.

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So now we have our ingroup from our limits and B for the X variable and we integrate the function with

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respect to X, and then we also integrate the function again with respect to Y between C and D to get

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the limits for C and D.

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So this joint probability equation here gives us the probability that the random variable X is between

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limits A and B, with the joint probability that Y is also between limits C and D.

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So this is just an extension of our equation to work out the probability from the density function using

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the integral.

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But we now we have to do a double integral because we got two parameters that we want to integrate over.

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We can also work out our marginal density functions so we have a joint probability density function

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for random variables, X and Y, if we integrated over the opposite variable, we can get the probability

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density function for the other variable.

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Sorry to work out our density function for a random variable X, we have to integrate our joint probability

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over Y and to do the opposite.

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If you want to work out our density function for the random variable, why we have to do our indefinite

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integral over the function with respect to X.

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We can also have a look at the expected value for when we're using a joint probability density function,

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so we recovered in the past that the expected value for when we're using a single random variable is

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just this equation shown here.

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So so we have a function of X and we want to work out the expected value of it.

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The answer this is just simply the integral of that function as a function of X multiplied by the probability

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density function of X integrated, all with respect to X.

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Now when we go to our multiple random variables, so now we just have to extend it by using a double

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integral over our X and Y.

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So it's pretty much the same equation, but now we're using a double integral for both parameters.

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So just like in basic probability, we can have independent random variables, and what this means is

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that if we have events and B, if they're independent, it means a total probability of event A and

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B happening.

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Is this the multiplication of the probability that A happens multiplied by the probability that Event

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B happens?

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So if they're independent, this relationship holds up.

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Now, a similar relationship can be found for the probability density functions for all random variables.

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So if we have a joint density function of X and Y, if events or the random variables X and Y are independent,

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it means that the density function is just going to be the same as multiplying the two individual marginal

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density functions together.

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So if this definition holds, it means that the random variables X and Y are independent of each other.

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So now let's have a look at the expected value for multiplication of independent random variables.

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So we want to work out the expected value when we multiply a random variables, X and Y together.

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So now using the equation that we showed on previous slide, we can work out our integral.

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So we just multiplying our random variables, X and Y.

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So that becomes a function and we multiply that by our density function so we can expand this equation

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out further.

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We actually know that our joint probability density function is independent, so we know that we can

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split it up into our F and Y random.

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So we know that we can split it up into our X and Y probability density functions.

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So we end up with this equation shown here.

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So we're basically just splitting up our joint density function into the two individual component density

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functions.

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So next we can move and separate out X and Y variables together so we can separate them out.

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And now we have two integrals multiplied together.

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So we have all the X terms in this integral and all Y terms in this integral area here.

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Now, if you have a look closer at these two integrals, you might look at them and they might seem

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pretty familiar.

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This is because this term here and this term here is simply just the expected values for the X and Y

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random variables.

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So using this here, we know that our definition, assuming independent expected value for random variables

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X multiplied, is just going to be the expected value of the X random variable multiplied by the expected

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value of the Y random variable.

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And of course, this only holds if X and Y are independent.

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We can also have a look at the expected value of the sum of independent random variables, so a function

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Z here is a function of two parameters, X and Y.

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And this function is just the sum of two independent functions, function G, which is just a function

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of X and function F and and function Hech, which is a function of Y.

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So we want to work out the expected value of our function Z and this just turns into our expected value

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of our function G plus our function H.

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So now riding out our expected value equation, we can pull in a function and put the function that

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we want to work at the expected value of in of you, multiply it by the probability density function.

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Now, we can expand this some out here into two terms.

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We can also expand out our probability density function.

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So if we assume X and Y are independent, we know that we can break it down into the two different individual

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probability density functions.

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So when we do that, we can get this equation here.

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So we split out our function of G into this one.

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A function of how much into this one over here.

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Multiply it by our two individual probability density functions.

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And like we did in the last example, now we can split this in this double integral into an integral

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respect to X and integral in respect to Y, because this is all just a single multiplication.

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So when we do that, we end up with this term here and we can also have a look at a different individual

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integrals and we might notice some commonalities between the two.

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Now, if you look at this integral here, this integral here is just going to equal to one, because

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this is just summing up the probability density function and we know that the probability density function

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has to equal one.

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We can do the same thing over here for X, that also has to equal one.

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So now we just end up with the sum of this integral and this integral and going back to our basic expectation

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in the integral, we can see that this integral here and this integral here are just the expected values

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for a function G and a function H.

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So this is basically saying that when we say this is saying that our expected value for the sum of two

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independent random variables is just the sum of the expected values for the two independent variables.

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So if we had our function, Z is equal to X plus, why then our expected value for this function is

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just going to be expected value for X plus expected value for Y.

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And again, this only holds when X and Y are independent.

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So these are two different identities for independent random variables that are useful to know.

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So now we can also look at dependent random variables, so the two random variables might not be independent.

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There might be some correlation between the two.

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So every time we query the joint custody function, we might end up with some correlated examples.

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So we can see on the right here, we have a graph that has the X random variable and Y random variable.

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And you can see that every time we create the density function, we might get different points that

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look like this.

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And it's pretty clear to see that these are correlated in some regard.

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We can see that there's basically the following in this case, a line of correlation.

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And this is actually a positive correlation because this means that as X gets bigger, Y gets bigger,

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you could have cases of negative correlation.

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So that would be on the other side when when it gets bigger, Y gets smaller.

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So be along with the opposite slope or you might have zero correlation.

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So there might be independent and that would just mean that random all over the place.

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There's no pattern to them.

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So to work out the correlation between the different components, we can look at the covariance.

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So in the past we looked at the variance of a parameter and that was just the expected value of the

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random variable, minus the main squared.

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So the covariance is very similar.

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But instead of being a function of a single random variable is now a function of two random variables.

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So here we have the expected value of the multiplication of our random variable X minus its main multiplied

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by a random variable, Y minus this mean.

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And we can expand this out and we can end up with this equation here.

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So so this is how we can work at the covariance coefficient.

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So we basically look at the expected value of X and Y minus the mean of X and Y.

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So using the covariance, we can work out a correlation coefficient.

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So that's basically this term here.

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So using the variance multiplied it by the standard deviation for the random variable X and Y gives

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us a correlation coefficient between negative one and one.

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Negative one means that negatively correlated positive one mean that positively correlated zero means

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are not correlated at all.

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So here is a summary of all the different concepts we have gone over in this video.

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We've talked about the joint probability density function and how to work out the probability from it.

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We've described how to work out the expected value equation for when we have a joint probability density

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function.

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We looked at the independence condition for a joint density function, just being the sum of the two

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individual density functions.

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We've looked at the concept of covariance.

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So how much?

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One parameter is correlated with another parameter.

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We've worked out how to work out a marginal density functions from our joint density function here and

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we've got a few independent expectation identities that will come in handy every now and again of that,

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the expected value of the sum that the expected value of the multiplication of random variables is just

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going to be expected.

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Value multiplied together and likewise with the sum is just going to be the sum of the expected values

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together.