For the random variables X and Y whose joint density function is given in Exercise 3.40 on page

step 1 For this problem, we have a joint density function. And we also are trying to find the convergen

For the random variables X and Y whose joint density...

Key Concepts

Integration in Probability

Integration in probability is the fundamental technique used to calculate expectations, variances, and probabilities for continuous random variables. It involves integrating the probability density functions (joint or marginal) over the domain of interest to obtain these statistical measures.

Covariance

Covariance quantifies how much two random variables change together and is defined as the difference between the expected value of their product and the product of their expected values, E[XY] - E[X]E[Y]. It is a key measure in probability and statistics to understand the linear relationship between two variables.

Expected Value

The expected value of a random variable is its long-run average or mean value, computed as a weighted average of all possible values the variable can take, where the weights are given by the probabilities of those values. For continuous random variables, this entails integrating the product of the variable and its probability density function over its support.

Joint Probability Density Function

The joint probability density function (pdf) describes the probability distribution of two continuous random variables simultaneously. It provides the basis for calculating joint probabilities and expectations for functions of these variables by integrating the joint pdf over the appropriate domains.

Marginal Density Function

The marginal density function is derived from the joint density function by integrating out the other variable. It gives the probability distribution of a single variable irrespective of the other, and is essential for determining the expected value and variance of that variable individually.

Transcript

00:01 For this problem, we have a joint density function.

00:04 And we also are trying to find the convergence.

00:13 So what we're going to do first is we're going to define what the convergence is.

00:18 So we're trying to find the convergence of x and y.

00:20 So it looks like this.

00:23 The conversions of x and y equal e of x of x and y.

00:50 Minus e of x and e of y.

00:58 And so the first thing that we're going to want to do is find e of x.

01:03 Then we're going to want to find e of y.

01:06 And so for this function, it's going to look a little different.

01:10 We're going to actually do g of x.

01:14 And g of x equals f of x, y, d, y.

01:20 So what this looks like is this.

01:23 G of x equals and we're just the reason why i said we do e of x first and the e of y because we're following the natural pattern of this problem but since it's a joint density problem it's going to look different and so it's going to have a function i mean it's going to have an integral and then it's going to have a function of x and y and so we're going to pluck it in and we have two thirds and then we have limits of 011 x plus 2 d y and then what that equals is this two thirds and then we have x 0 and 1 d y plus 2 limits of 0 and 1 y d y and so now what we're going to want to do is we're going to want to evaluate that and that is 2 3 3 3 3 of x plus 2 1 1 1 1 over two and you have your limits of one and zero and what this comes out to be is two -thirds and then you have in parentheses x plus one and so what we are defining what g of x is is its marginal density so now we're gonna find the expected value of x which is e of x and it's e of capital x which it makes a different it does and so we'll put it in a different color and it's e of x and it's a limit of 0 1 and it's x times g of x d x which means we're going to do some type of antiderivative and so let's just plug in the problem and work it out it equals two -thirds and then we have a limit of 1 to 0 and then inside there we have x squared plus x dx and what this equals this is two -thirds we have this limit of one to zero you're going to have x squared dx plus two -third one to zero x d x dx and this equals five nths so we're going to find the marginal density of h of y and then we'll find the expected value of y which is e of y and so let's start with the marginal density so it is denoted as h and y equals f of some magnitude of x and then you have f of x y dx and that's two thirds of x dx plus two y d x x and so so what we do from there is we do two thirds...

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