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Now that we've been talking about multiple random variable is happening at the same time, we can generalize

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the single random variable into a vector form.

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So now we have a random vector and a basically a random vector is just a vector where each element inside

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the vector is its own random variable.

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So, for example, here we have a random vector X and we have random X1 x2 all the way up to X and all.

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We have this random vector here Y and that has random variable is Y one y two all the way down to Y.

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And so this is just a mathematical way of writing groups of random variables together in vector form.

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Now, the main of the random vectors can be calculated in the same way as a single random variable,

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but it's done on per element basis.

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So if we have a random vector X here, the main vector or the expected value of this random vector X

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is simply just the expected value of each of the different random variables individually.

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We can also work out the covariance so the covariance between two random variable vectors can be calculated,

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which forms a covariance matrix.

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So instead of a single value is now a matrix.

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So let's say we have a random variable vector X and Y where we have the random variables X one all the

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way to the end and Y one all the way to Y, and we can work out the covariance matrix using the expected

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value of our X random variable vector minus the X main multiplied by Y random variable minus Y mean

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transpose.

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And again we can simplify this down into this equation here.

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So when we do this operation here, we end up with a covariance matrix.

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So each element inside this covariance matrix is a covariance value for each of us ran a very, very

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each of those random variables with respect to a different random variable.

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So this term here is a covariance or random variable, X one with Y one.

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Here is a variance on X one with Y two, likewise over across four X one all the way up to M and down.

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The Matrix is similar all the way down to extend and Y m.

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Since each element in the random variable Vector X itself is a random variable, it is possible to calculate

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the covariance of each element in the vector with respect to each other element in the vector.

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And when we do this, we get the autocorrelation matrix.

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So this is just the covariance matrix for the random vector and itself.

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So here we are working out the expected value of our X random variable vector minus its main multiplied

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by the same thing transpose.

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And this gives us the covariance matrix of our random variable vector X.

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And if you have a look at what's contained inside this covariance matrix, it becomes a variance and

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cross covariance, which described the probability density function.

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So on the diagonal here we have the variances.

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So Sigma squared for X1 all the way down to our sigma squared X.

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And so these are the variances for the different random variables and our of diagonal terms.

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Here are the code, the crosscourt variances.

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So these turn to these talk about how correlated the different variables are with respect to each other.

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So here we have the cross covariance between random variable X1 and random variable x2.

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And down here you can see that we have a mirror image.

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We have the cross covariance of random variable x2 with respect to x1 the of diagonal terms, you are

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going to be symmetrical and are going to be replicated.

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Now if we look at the properties of the covariance matrix, all the autocorrelation matrix, the covariance

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matrix is symmetrical.

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So this is basically because our correlation or across the variances for IJA is the same as the variance

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for Ajai because it's the same random variables.

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This means that overall our covariance matrix, our C X is equal to, say, X transpose, because if

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we transpose this matrix, the diagonal matrix stays the same and the terms flip around.

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It become the mirror image of each other.

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But we know that this time here is equal to this time here.

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Likewise, this time here is equal to this time here.

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So we flip around the diagonal, we end up with the same matrix.

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And we also know that our covariance matrixes square because it is and by n matrix.

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Now, for a convergence matrix to be valid, it also has to be positive, semi definite.

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So this just means that if we multiply our variance matrix by any vectors that here transpose times

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by Z, the resulting operation is always going to be greater than or equal to zero.

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And this means the matrix is positive.

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So this is important later on.

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So any probability distribution that has a covariance magic that's positive, some definite is valid.

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If it's not positive, something definite and it's not symmetrical, it's not a valid distribution for

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probability density function.