- #1
Saladsamurai
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Homework Statement
I am having trouble following what is going on in this solution. We are looking to find the expectation value of:
[tex]f(x,y)=\frac{1}{4^{x+y}}\cdot\frac{9}{16}[/tex]
I have gotten it down to:
[tex]E(X) = \frac{3}{4}\sum_{x=0}^\infty x\cdot\left(\frac{1}{4}\right)^x\qquad(1)[/tex]
We know that for a geometric series with an initial value of 1 we can write for 0 < r < 1:
[tex]\sum_{x=0}^\infty r^x = \frac{1}{1-r}[/tex]
taking the derivative of both sides wrt 'r' yields:
[tex]\sum_{x=1}^\infty r^{x-1} = \frac{1}{(1-r)^2}\qquad(2)[/tex]Here is where I get confused:
I thought it was a simple matter of plugging in:
[tex]\frac{3}{4}\cdot\frac{1}{(1-1/4)^2} = 4/3[/tex]
However, the solution gives:[tex]\frac{3}{4}\cdot\underbrace{\frac{1}{4}}\cdot\frac{1}{(1-1/4)^2} = 1/3[/tex]
I am a little confused as to where the factor of 1/4 is coming from. I am having a feeling that it has something to do with the fact that (1) runs from 0 to infinite and (2) runs from 1 to infinite.
Any thoughts?
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