Find the constant that makes f(x,y) a PDF

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Homework Help Overview

The discussion revolves around finding the constant k that makes a given function f(x,y) a probability density function (PDF). The joint density function is defined as f(x,y) = e^(-kX) for specified ranges of X and Y, with k being an unknown constant.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the criteria for a function to qualify as a PDF, specifically focusing on the requirement that the integral of the joint PDF equals 1. Questions are raised about the independence of X and Y and the implications of this on the problem.

Discussion Status

Some participants have provided guidance on the integral approach necessary to determine the value of k. There is acknowledgment of uncertainty regarding the correctness of the derived value for k, and the discussion reflects a mix of exploration and attempts to clarify the problem setup.

Contextual Notes

There is a lack of clarity regarding the independence of X and Y, which is noted as a separate part of the question. Additionally, participants express uncertainty about the initial steps to take in solving the problem.

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Homework Statement


Find the value of k that makes this a probability density function.
The question does not specify whether X and Y are independent or dependent. That is actually another part this question.

Homework Equations


Let X and Y have a joint density function given by
f(x,y) =e^(-kX), 0<=X<inf, 0<=Y<=1,
k is a constant, and 0 elsewhere

The Attempt at a Solution


I need quite a bit of direction with this question, I have no idea where to start.
 
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should I be posting this question in the statistics section of pf?
 
alias said:

Homework Statement


Find the value of k that makes this a probability density function.
The question does not specify whether X and Y are independent or dependent. That is actually another part this question.

Homework Equations


Let X and Y have a joint density function given by
f(x,y) =e^(-kX), 0<=X<inf, 0<=Y<=1,
k is a constant, and 0 elsewhere

The Attempt at a Solution


I need quite a bit of direction with this question, I have no idea where to start.
This is the right place for this question.

How can you tell whether a function is a "probability density function"?
 
The integral of a joint PDF = 1:
f(x,y) dxdy = 1
Sorry I don't have a better response.
 
That's the direction I was looking for.

For a PDF, the integral should be 1, right?

What does k need to be so that this integral is 1?
\int_{y = 0}^1 \int_{x = 0}^{\infty} e^{-kx} dx~dy = 1
 
I end up with k=1, still not sure if I'm right...
 
got it, thanks.
 

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