Find the covariance of the random variables X and Y of Exercise 3.44 on page 125 .

Name: Problem 46, Chapter 4: Probability and Statistics for Engineers and Scientists
Uploaded: 2021-06-16T00:08:37-08:00
Duration: 13 min 57 s
Channel: Tatiana Graham
Description: Find the covariance of the random variables X and Y of Exercise 3.44 on page 125 .

Find the covariance of the random variables X and Y of...

Key Concepts

Covariance

Covariance measures the degree to which two random variables change together. Defined as E[(X - E[X])(Y - E[Y])], it shows whether increases in one variable generally correspond to increases in the other (positive covariance) or to decreases (negative covariance). It is a fundamental tool for understanding relationships between variables in probability and statistics.

Expected Value

The expected value is the mean or average of a random variable, representing the long-run outcome of a probabilistic process. It is crucial for calculating covariance, as the definition relies on the differences between each random variable and its respective expected value.

Joint Distribution

The joint distribution of two random variables describes the probability of different pairs of outcomes occurring together. Understanding the joint distribution is essential when computing combined statistics like the expectation of the product of two variables (E[XY]), which is a key component of the covariance formula.

Linear Relationships

Linear relationships between random variables reflect how one variable may change in a consistent proportion with another. Covariance provides insight into the direction of this relationship, and while it indicates whether the variables tend to move together, it does not offer a scale-invariant measure of the strength of the relationship, which is addressed by the correlation coefficient.

Transcript

00:01 So for this problem, we are going to do a lot of different things, but our main goal is to find the covariance of the variable x and y.

00:11 But like i said, it is going to take a lot of steps.

00:15 So let's get started.

00:17 So x is denoted as a pressure of a right tire.

00:23 And y is denoted as the pressure of a left tire.

00:25 So x and y is the joint density.

00:31 That we're given.

00:32 So for this problem, we have f of x and y, and it's k and it's x squared plus y squared, and then we have zero.

00:55 And these are our two piecewise functions and this is otherwise.

01:01 So let's look at our limits for this part that is very important for x and y our limits.

01:10 So it's 30 is, so x is in the middle of these two points.

01:21 So x can be less than or greater than 30, less than 30, less than equal to 30 or greater than 50 or, oh, i'm sorry.

01:33 30 is less than x and x is greater than 50.

01:38 And then for y, we have 30 is less.

01:44 Than or greater or equal to y and y is greater than is less than 50 sorry so what that means is is we need to find the value of k because we know that k exists k has to exist for this to even work and so what we'll do is is we're going to kind of tweak together our own little equation.

02:20 So we have x and y integrals and we have f of x and y, dy, dx equals 1.

02:33 And the reason why we know this is because given a pdf is even valid or is even possible, it has to equal one.

02:43 That's how we know that we have a random variable.

02:46 And so that is our only parameter that.

02:49 That we're working with.

02:50 So let's kind of work this out, what this actually means.

02:56 So we have these limits for x and y, and then we have k, and then we have x squared plus y squared, d y, d x, and we'll integrate from here, and we'll put what we're gonna kick k out, which is our constant.

03:34 And we have our first integration.

03:57 And this is our first integration.

03:59 And so what it is is we have x squared plus y squared over three.

04:07 And that's our first antiderivative that we did.

04:11 So next, we're going to evaluate our limits.

04:23 So we have k.

04:28 And we have our limits of 50 and 30.

04:31 And we have 20x squared plus 9 ,800 ,000 over 3...