Expected value of X and Y, E[XY] for uniform random variables

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Homework Help Overview

The discussion revolves around the covariance of two random variables, X and Y, where X is uniformly distributed between -1 and 1, and Y is defined as the square of X. Participants are exploring the possibility of determining the covariance, cov(X, Y), and the expected value E[XY].

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are considering the relationship between Y and X, particularly how Y can be expressed in terms of X. There are questions about how to compute E[XY] and whether it can be simplified using the definition of Y.

Discussion Status

Some participants have suggested substituting Y with X^2 in the equations, indicating a potential direction for simplifying the problem. However, there is still uncertainty about the implications of this substitution and how it affects the calculation of expected values.

Contextual Notes

There is an ongoing discussion about the definitions and relationships between the variables, particularly regarding the distributions and the nature of Y as a function of X. The original poster's attempt at a solution indicates a need for further clarification on the integration process involved in finding E[XY].

Dustinsfl
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Homework Statement


If ##X\sim\mathcal{U}(-1,1)## and ##Y = X^2##, is it possible to determine to ##cov(X, Y)##?

Homework Equations


\begin{align}
f_x &=
\begin{cases}
1/2, & -1<x<1\\
0, & \text{otherwise}
\end{cases}\\
f_y &=
\begin{cases}
1/\sqrt{y}, & 0<x<1\\
0, & \text{otherwise}
\end{cases}
\end{align}

The Attempt at a Solution


$$
cov(X,Y) = E[XY] - E[X]E[Y] = E[XY] - 0\cdot 1/2 = E[XY]
$$
Now
$$
E[XY] = \int_0^1\int_{-1}^1g(X, Y)f_{x,y}(x,y)dxdy
$$
From the information that I have, can I determine ##E[XY]##?
 
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I suggest using the fact that ##Y = X^2##...
 
Orodruin said:
I suggest using the fact that ##Y = X^2##...

How?
 
What do you get if you replace ##Y## everywhere by ##X^2##?
 
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Likes   Reactions: Dustinsfl
Dustinsfl said:
How?

For any observed value ##X = x##, the observed (or, rather, computed) value of ##Y## is ##y = x^2##. It not just that ##Y## and ##X^2## have the same distribution; much more than that is true: ##Y## IS ##X^2##.
 

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