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Bivariate Transformation of Random Variables

  • Thread starter Yagoda
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  • #1
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Homework Statement


Two RVs X1 and X2 are continuous and have joint pdf [itex]
f_{X_1,X_2}(x_1, x_2) = \begin{cases} x_1+x_2 &\mbox{for } 0 < x_1 < 1; 0 < x_2 < 1
\\
0 & \mbox{ } \text{otherwise}. \end{cases} [/itex]

Find the pdf of [itex]Y = X_1X_2[/itex].


Homework Equations


I'm using the transformation "shortcut' that says if U = g1(x,y), V = g2(x,y) and h1(u,v) = x, h2(u,v) = y then
[itex]f_{U,V}(u,v) = f_{X,Y}(h_1(u,v), h_2(u,v) \times |J| [/itex] where |J| is the absolute value of the determinant of the Jacobian of h1 and h2.


The Attempt at a Solution


In this setting I let [itex]Y = X_1X_2[/itex] and [itex]Z = X_1[/itex] so [itex]h_1(y,z) = Z[/itex] and [itex]h_2(y,z) = Y/Z[/itex]. Then I calculated [itex]|J| = \frac{1}{Z}[/itex] so plugging in I got that
[itex]f_{Y,Z}(y,z) = (z + \frac{y}{z})(\frac{1}{z}) = 1 + \frac{y}{z^2}[/itex] where [itex]0 < z < 1, 0 < y < z[/itex].
So to find the pdf of Y, integrate with respect to z over all possible values of z, but this integral does not converge when evaluated from 0 to 1.

I've tried other choices for Z, but I've ended up with the same problem.
 
Last edited:

Answers and Replies

  • #2
statdad
Homework Helper
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Note: if both [itex] 0 \le x_1 \le 1 [/itex] and [itex] 0 \le x_2 \le 1 [/itex], how does the product
[itex] X_1 X_2 [/itex] compare in size to [itex] X_1 [/itex] alone?

Edited to add: I have not stepped through all of your work to simplify the transformation, simply noted that you have [itex] 0 < z < 1 [/itex] and [itex] 0 < y < 1 [/itex].
 
  • #3
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Since [itex]X_1[/itex] and [itex]X_2[/itex] are both smaller than 1, then their product will be smaller than [itex]X_1[/itex] alone, but I'm not seeing how this helps me find the pdf of [itex]Y[/itex].

Are the limits [itex]0 < y < z[/itex] not correct? It seems that this needs to be true in order for [itex]Y/Z[/itex] to be less than 1?

Now that I'm looking at it again, is this transformation shortcut even valid in this situation? I'm not sure if my [itex]h_1, h_2[/itex] are workable in this case.
 

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