Find the covariance of the random variables X and Y of Exercise 3.39 on page 125 .

Name: Problem 44, Chapter 4: Probability and Statistics for Engineers and Scientists
Uploaded: 2021-06-15T23:47:56-08:00
Duration: 8 min 36 s
Channel: Tatiana Graham
Description: Find the covariance of the random variables X and Y of Exercise 3.39 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 question, we are trying to find the convergence of x and y when we have those two random variables.

00:09 So the first thing we're going to do is we have a given joint distribution, which i'll write here.

00:17 And what it is is if you are fuzzy of joint distribution, it's not a part of this problem you're giving it, but i'll show you how to do it.

00:25 So when you have a function x and y, you have a table.

00:39 And you've x here and you have y here and for our x's we have zero one and two and for our y's you have zero one two and three we have g of x here and g of x is a sum of all of our x's and i'm sorry i have these in the just map x is right there because we're going to have all of our sum of all our x is here and y we'll just go here and then we'll have h of y which will be all the sum of our y's and whenever you're confused on something like that you can always go back and just look at the map on the chart and so where you see here this is what the completed chart looks like you always know that you're right is if your g of x's and your sums of all your h of y's if they all equal one together you're you're you're all you're and the reason is because we have a pdf rule that says that all of our joint distributions always have to equal one to be a valid random variable.

02:11 So that's how we kind of know that we're right.

02:15 And now what we're going to do is we're going to move to solving the mean.

02:21 And in the next three steps, we're going to work in three steps.

02:24 We're going to solve the mean, then we're going to solve the mean of y.

02:31 So we'll start with the mean of x, then we'll start with the mean of y.

02:35 And then we're going to solve the double summation of x and y.

02:42 So we're going to solve the e of x of y, and then that's from there.

02:47 We're going to be able to find the covariance of x and y.

02:51 So we start with the mean of x.

02:58 And just so you know for variance problems, the mean of x is equal to e of x, our second term in the covariance formula.

03:10 And so for this problem, when we have, you can think about this, is three variables.

03:17 And x stark said zero.