- #1

Master1022

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- 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.

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) ##.

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.

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.