Show that Cov(X, Y)=E(X Y)-E(X) E(Y), and use this result to conclude that Cov(X, Y)=0 if X and

step 1 For this problem, we're looking at covariance. It's a definition of covariance of two events, x

Show that Cov(X, Y)=E(X Y)-E(X) E(Y), and use this result to...

Key Concepts

Covariance

Covariance is a measure of how much two random variables change together. It is defined as the expected value of the product of the deviations of each variable from their respective means. The formula Cov(X, Y) = E(XY) - E(X)E(Y) captures this idea by comparing the actual product of the variables to what would be expected if they were independent. This concept is key in statistics for understanding the direction of the linear relationship between variables.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency. It is calculated as the weighted average of all possible outcomes, where each outcome is weighted by its probability. In the context of covariance, the expected value provides a baseline (the mean) from which deviations are measured to determine how the variables are related.

Independence of Random Variables

Independence between random variables indicates that the occurrence or the value of one variable does not affect the occurrence or value of the other. A vital property of independent random variables is that the expected value of their product equals the product of their expected values, i.e., E(XY) = E(X)E(Y). This result is crucial when using the covariance formula since it implies that independent variables will have a covariance of zero, indicating no linear relationship between them.

Linearity of Expectation

Linearity of expectation is the property that the expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether the random variables are independent. This property is essential in the derivation of covariance and other statistical measures. It allows the separation of complex expected value calculations into simpler, more manageable parts.

Transcript

00:01 For this problem, we're looking at covariance.

00:04 It's a definition of covariance of two events, x and y, is equal to the expected value of xy minus the expected value of x, times the expected value of y.

00:45 Okay, so this is what we want to show.

00:47 So i'll start with the definition of covariance, which is the expected value of x minus the expected value of x, y minus the expected value of y, okay, so the first thing we're going to do is distribute.

01:27 Okay, so we're just going to sort of multiply everything that's in here.

01:31 So we're going to get the expected value, and so we'll have our sort of first, outside, inside, last, so, we get the expected value of x, y, minus x times the expected value of y, minus y times the expected value of x, plus the expected value of x, plus the expected value of x times the expected value of y.

02:15 And then we're going to use the property of expected value, and so that just says that the expected value of the sum, is equal to the sum of expected values.

02:29 So that means that this is going to be equal to the expected value of x times y minus the expected value of x times the expected value of y, minus the expected value of y times the expected value of x plus the expected value of the expected value of the product of the expected value of x and expected value of y plus b is equal to the expected value, well, a times the expected value of x plus b.

03:59 Okay, so we can use the fact that the expected value of x and an expected value of y are constants here and pull some of these pieces, to use them apart a little more.

04:11 So we're going to get that the expected value of x times y minus then the expected value of x times the expected value of y minus the expected value of y times the expected value of x plus the expected value of x times the expected value of x times the expected value of y okay, and then we kind of do the same thing with this piece here to get that to simplify as well.

05:30 So then that's just going to end up being the expected value of x times the expected value of y for the final expression here...

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