Probability: Multivariate distribution change of variables

In summary, the conversation discussed finding the probability distribution of U_1, given Y_1 and Y_2 with a joint pdf of 8y_1 y_2 for 0 < y_1 < y_2 < 1. The process involved defining U_2 as Y_2 and using transformations to find the marginal distribution of U_1. The limits for u_2 were determined to be 0 and 1, and the possible values for u_1 were found to be between 0 and 1. It was determined that this also implies the limits for u_2 to be between 0 and 1.
  • #1
Master1022
611
117
Homework Statement
Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf:
[tex] f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 [/tex] for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ## p(u_1) ##.
Relevant Equations
Jacobian
Hi,

I was attempting the problem above and got stuck along the way.

Problem:
Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf:
[tex] f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 [/tex] for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ## p(u_1) ##.

Attempt:
We are not given a ## U_2 ##, but the problem provides a hint that we can define an arbitrary value for ## U_2 ##, for example, ## Y_2 ##. Then we can use that to find ## f(u_1, u_2) ## and then integrate with respect to ## u_2 ## to get ## f(u_1)##. It is the final step where I am confused as I am not completely sure about the limits for ## u_2 ##.

The working is as follows:

1. Define ## U_2 = Y_2 ##. Both transformations are one-to-one transformations so no extra steps are needed.

2. Find y1 and y2 in terms of u1 and u2. This yields ## y_1 = u_1 u_2 ## and ## y_2 = u_2 ##

3. Find the magnitude of the Jacobian, which turns out to be ## |J| = |u_2| ##

4. Find ## f_{u_1, u_2} (u_1, u_2) = f_{y_1, y_2} (y_1, y_2)|J| = 8u_1 u_2 ^3 ##

5. Then we can integrate to find the marginal distribution of ## u_1 ##. We need to use the inequality ## 0 < y_1 < y_2 < 1 ## to find the limits. Splitting it up we get ## u_1 u_2 > 0 ## and ## u_1 u_2 < u_2 < 1 ##. At this point, I am not quite sure how to use the inequalities. Should I just be using the limits ## 0 ## to ## 1 ##? I think I may be overthinking it as I seem to think there should be some use of ## u_1 ##...

Any help would be greatly appreciated.
 
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  • #2
To find the marginal distribution of [itex]U_1[/itex], you integrate with respect to [itex]u_2[/itex]. Since you have taken [itex]u_2 = y_2[/itex] its limits are indeed 0 and 1.

From [itex]0 < y_1 < y_2[/itex] you can determine the possible values of [itex]U_1[/itex].
 
  • #3
pasmith said:
To find the marginal distribution of [itex]U_1[/itex], you integrate with respect to [itex]u_2[/itex]. Since you have taken [itex]u_2 = y_2[/itex] its limits are indeed 0 and 1.

You also have from [itex]0 < y_1 < y_2[/itex] that the possible values of [itex]u_1 y_1/y_2[/itex] lie in [itex](0,1)[/itex].
Thank you @pasmith ! So that means that ## 0 < u_1 < 1 ##. Am I correct in thinking that this implies that the inequality leads to ## 0 < u_2 < 1 ## as well?
 

Related to Probability: Multivariate distribution change of variables

1. What is a multivariate distribution?

A multivariate distribution is a probability distribution that involves more than one random variable. This means that it describes the probabilities of multiple events occurring simultaneously.

2. What is a change of variables in probability?

A change of variables in probability refers to the process of transforming the original random variables in a multivariate distribution into new variables. This is often done to simplify the calculation of probabilities or to better understand the relationship between the variables.

3. Why is it important to understand the change of variables in probability?

Understanding the change of variables in probability is important because it allows us to analyze and interpret complex multivariate distributions. It also helps us to make predictions and decisions based on the relationships between the variables.

4. How is the change of variables related to the concept of independence?

The change of variables is related to the concept of independence because it can be used to determine if two or more variables in a multivariate distribution are independent. If the new variables are independent, then the original variables are also independent.

5. What are some common techniques for handling multivariate distributions with change of variables?

Some common techniques for handling multivariate distributions with change of variables include the Jacobian transformation, the inverse transformation method, and the moment generating function. These techniques allow us to calculate probabilities and make inferences about the variables in the multivariate distribution.

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