Joint problem density function problem

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Discussion Overview

The discussion revolves around a joint probability density function problem involving the function f(x,y) = e^{-x} for the range 0 ≤ x ≤ y ≤ ∞. Participants are trying to understand how to determine the range and calculate the probability P(x+y≤1).

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about forming the range for the given problem and attempts to integrate over a proposed range.
  • Another participant points out that the density function appears to depend on both x and y, despite being expressed as a function of x only.
  • A correction is made regarding the range, suggesting it should be 0 ≤ y ≤ x ≤ ∞ instead of the initially stated range.
  • There is a discussion about determining the value of A by ensuring the total probability integrates to 1, with one participant providing an equation for A.
  • Another participant questions whether it will be possible to find P(x+y≤1) after determining A.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct range for the variables or the clarity of the problem statement. Multiple competing views regarding the interpretation of the density function and the integration limits remain unresolved.

Contextual Notes

There are limitations in the clarity of the problem statement and the assumptions regarding the range of integration. The dependence on the variable A and the conditions for the probability calculations are also not fully resolved.

Lewis7879
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I need help guys I can't understand this
Can anyone explain thoroughly how do I form the range for this question?
f(x,y)= e-x for 0≤x≤y≤∞
0 Otherwise

Find P(x+y≤1)
I attempted this by integrating through the range of
0≤y≤(1-x) and 0≤x≤∞ but that doesn't seem right
 
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The statement is confusing. You appear to have a density function in x and y, which is a function of x only.
 
mathman said:
The statement is confusing. You appear to have a density function in x and y, which is a function of x only.
Hello mathman there's a slight error with range I made in the question which is 0≤y≤x≤∞
There was no other problems with the question as I was asked this way.
 
No, f(x,y)= e^{-x} for 0\le x\le y\le A is a function of both x and y. To determine "A", use the fact that the "total" probability must be 1:
\int_{y= 0}^A\int_{x= 0}^y e^{-x} dx dy= 1
 
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Getting an equation for A is easy enough. A+e^{-A}=2. I am confused as to what is the question.
 
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HallsofIvy said:
No, f(x,y)= e^{-x} for 0\le x\le y\le A is a function of both x and y. To determine "A", use the fact that the "total" probability must be 1:
\int_{y= 0}^A\int_{x= 0}^y e^{-x} dx dy= 1
Will I be able to find P(x+y≤1) after determine A?
 

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