The Cov(X,Y) on a square dart board

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Discussion Overview

The discussion revolves around the computation of the covariance Cov(X, Y) for random variables X and Y, which represent points picked uniformly from a unit square dart board. Participants explore the expectations E(X), E(Y), and E(XY), and discuss the implications of their calculations.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant computes E(X) and E(Y) as 0.5 using the integral xf(x), but struggles with calculating E(XY).
  • Another participant suggests that E(XY) can be computed using a double integral ∫∫xy, questioning the initial participant's approach.
  • A later reply indicates that the computed covariance of zero raises concerns about potential errors, while also suggesting that it may imply independence between X and Y.
  • One participant notes that a covariance of zero is a common outcome in textbook examples.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the calculation of E(XY) and the implications of a zero covariance. There is no consensus on the correctness of the calculations or the interpretation of the results.

Contextual Notes

Participants do not clarify the definitions or assumptions underlying their calculations, and the discussion lacks resolution on the accuracy of the computed expectations and covariance.

BookMark440
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A point (X, Y) is picked at random, uniformly from the square with corners at (0,0), (1,0), (0,1) and (1,1). Compute Cov{X, Y}.

I think of this as darts thrown at a unit square dart board.

Cov(X,Y) = E(XY) - E(X)E(Y).

I compute that E(X)=E(Y)= 0.5 using the integral xf(x).

But I cannot figure out how to approach computing E(XY). Or is there a better strategy for solving this?

THANKS!
 
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Welcome to PF!

BookMark440 said:
I compute that E(X)=E(Y)= 0.5 using the integral xf(x).

But I cannot figure out how to approach computing E(XY).

Hi BookMark440! Welcome to PF! :smile:

(I'm not sure exactly what you mean by f(x))

If you got E(X) from ∫x, why can't you get E(XY) from ∫∫xy ?
 
Thanks. I did that and ended up getting zero for an answer, which always makes me think I made an error. In this case, I guess it just verifies that X,Y are independent.
 
A covariance of zero is very common in textbook examples.
 

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