Finding Density Functions for Randomly Chosen Points in a Unit Square

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Homework Help Overview

The problem involves determining the density functions for the sum and product of the coordinates of a randomly chosen point within a unit square. The context is rooted in probability and geometry.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between choosing a point in the unit square and independent uniform random variables. There is a focus on finding the density function for the sum of the coordinates, with some questioning the final steps needed for calculation.

Discussion Status

Some participants have offered insights into the geometric interpretation of the problem, suggesting that drawing lines of constant x+y may assist in finding the density function. Multiple interpretations of the problem are being explored, but no consensus has been reached.

Contextual Notes

There is an emphasis on using geometric methods to derive the density functions, and participants are navigating the constraints of the problem without fully resolving the calculations.

Fuquan22
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Homework Statement


A point Q is chosen at random inside the unit square. What is the density function of the sum of the coordinates of point Q? What is the density function of the product of the coordinates of the point Q? Use geometry to find these densities.


Homework Equations


P(a < X < b) = Integral (a,b) of f(x)dx


The Attempt at a Solution


I know that the interval for the first part has to be between 0 and 2, but I don't know f(x)
 
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choosing a point at random from unit square is in effect equivalent to choosing 2 independent uniform random variables, say X & Y, on the interval [0,1].

Then you want to find the density function for Z = X + Y
 
Last edited:
Yes but how do i go about finding that density function? I just don't know the final step to calculating a density function.
 
It says use geometry. So does drawing lines of constant x+y help you solve it?
 

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