(a) Compute the covariance between X and Y in Exercise 11 (b) Compute the correlation coefficien

(a) Compute the covariance between X and Y in Exercise 11...

Key Concepts

Covariance

Covariance is a statistical measure that indicates the direction of the linear relationship between two random variables. It is calculated as the expected value of the product of the deviations of each variable from their respective means. A positive covariance indicates that the variables tend to increase or decrease together, while a negative covariance suggests that as one variable increases, the other tends to decrease.

Correlation Coefficient

The correlation coefficient is a normalized measure of the strength and direction of the linear relationship between two random variables. It is obtained by dividing the covariance of the variables by the product of their standard deviations, yielding a value that ranges from -1 to 1. This coefficient allows for comparisons across different pairs of variables, with values close to 1 or -1 indicating a strong linear relationship and values near 0 indicating a weak relationship.

Transcript

00:01 In the import a, we're asked to compute the covariance for x and y in exercise 11.

00:22 Looking at exercise 11 in this chapter, we have at the covariance, we found first we find the the expected value of x.

00:34 So if it's at the value of x, this is the integral from 20 to 30 of x times fx of x.

00:49 We found that fx of x was 10 kx squared plus 0 .05, so this is the same as the interval between the 30 of x times 10 k x squared plus 1 .5, bx.

01:17 We also found k previously in that problem.

01:23 In the simple rule is a little bit lengthy, but it is easy to carry out.

01:29 And the angle calculate the exact value 1925 over 76 and as a decimal this is 25 .3 to 9 approximately.

02:00 Now looking at this region it is a rectangle.

02:08 So the symmetry we have that expected value of y is the same as the expected value of x.

02:25 The value of y, mostly approximately, as .5 .329.

02:42 Now we want to calculate the value of x times y.

02:51 We know the nine dependence, so you can't just break this up.

02:57 So this is the integral from .30 again of x times y times f of x y times f of x y.

03:16 Be y, d x, and we have that f of xy, and that previous problem, is kx squared, k times x squared plus y squared, the x, y, and once again, it's not impossible to evaluate this integral, but it is mighty.

03:50 After doing so, you'll catch it something similar to 24 ,375, over 38, which has a decimal, which is approximately 641 .447.

04:25 So, call this together, we have the covariance of x and y...

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