Applied Probability Calculation not Working Out (Integrals)

In summary, the joint density of X and Y is given by f(x,y) = [e-x/y)e-y ] / y, where x is between 0 and infinity and y is between 0 and infinity. The task is to find E[X|Y=y] = y, which requires finding fX|Y(x|y) by dividing f(x,y) by fY(y) and integrating with respect to x. The correct integral is from 0 to infinity, which results in a value of y. The solution in the manual is incorrect.
  • #1
jumbogala
423
4

Homework Statement


The joint density of X and Y is given by
f(x,y) = [e-x/y)e-y ] / y

x is between 0 and infinity; y is between 0 and infinity

Show E[X |Y = y] = y.

Jump to the last part of this post in bold if you just want to check my calculus calculations and skip the probability stuff.

Homework Equations


The Attempt at a Solution


First I need to find fX|Y (x|y). To do this I need to take f(x,y) and divide by fY(y).

I think it's the latter which is messing me up. My understanding is that to find it, I need to integrate f(x,y) with respect to x.

The solutions manual does that, but there are no limits on their integral. Shouldn't the limits be from 0 to infinity?

They have [(1/y)e-x/ye-y ] / [e-y ∫ (1/y)e(-x/y) dx ] = (1/y)e-x/y

Which in my calculations doesn't work out! I get -1/y. What am I doing wrong?
 
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  • #2
jumbogala said:
The solutions manual does that, but there are no limits on their integral. Shouldn't the limits be from 0 to infinity?

They have [(1/y)e-x/ye-y ] / [e-y ∫ (1/y)e(-x/y) dx ] = (1/y)e-x/y

Which in my calculations doesn't work out! I get -1/y. What am I doing wrong?

The integral is

[tex] \int_0^\infty e^{-x/y} dx = -y ( 0 - 1 ) = y.[/tex]
 

FAQ: Applied Probability Calculation not Working Out (Integrals)

1. Why is my applied probability calculation not working out?

There could be several reasons why your applied probability calculation is not working out. It could be due to errors in your input data, incorrect assumptions or simplifications in your model, or issues with the integration method used. It is important to carefully check your calculations and assumptions to identify the source of the problem.

2. How can I improve the accuracy of my applied probability calculation?

To improve the accuracy of your applied probability calculation, you can try using more advanced integration methods such as Monte Carlo simulation or numerical integration techniques. Additionally, double-checking your input data and assumptions can also help improve the accuracy of your calculation.

3. What are some common mistakes to avoid when performing applied probability calculations?

One common mistake to avoid is using oversimplified or unrealistic assumptions in your model. This can lead to inaccurate results. It is also important to double-check your input data and ensure that you are using the correct units and variables in your calculation.

4. Can I use applied probability calculations in real-world applications?

Yes, applied probability calculations are often used in real-world applications to make informed decisions and predictions. They can be used in fields such as finance, engineering, and medicine to assess risk and make data-driven decisions.

5. Are there any resources or tools that can help with applied probability calculations?

Yes, there are many resources and tools available to assist with applied probability calculations. These include software packages, online calculators, and textbooks or tutorials on the subject. It is also helpful to consult with other experts or professionals in the field for guidance and advice.

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