Expected values of random variables !

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I don't completely understand why the area of the proof circled in red is true.

Any advice would be appreciated.

https://dl.dropboxusercontent.com/u/33103477/Q1.jpg [Broken]
 
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Office_Shredder

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X1 is a random variable whose mean is [itex] \mu[/itex] by definition. Can you elaborate on your confusion?
 
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X1 is a random variable whose mean is [itex] \mu[/itex] by definition. Can you elaborate on your confusion?
Where is this defined ? Is is part of the definition of 'Expectation' ?
 

Office_Shredder

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Why don't you tell us what you think X1 is, and what [itex] \mu [/itex] is, and we can work from there.
 
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Why don't you tell us what you think X1 is, and what [itex] \mu [/itex] is, and we can work from there.
μ=[itex]\frac{\sum X_i}{N}[/itex]

[itex]x_1[/itex] is just a variable
 

Office_Shredder

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μ=[itex]\frac{\sum X_i}{N}[/itex]
No, the thing on the right hand side is [itex] \overline{X} [/itex], not [itex] \mu [/itex]. To give an example, suppose I flip ten coins, and assign a value of 1 to a heads, and 0 to a tails. I might get the following:

1,0,0,1,0,0,1,0,1,0.

[itex] \mu[/itex] in this context is the expected value of a single flip of the coin, which is .5. [itex]\overline{X}[/itex] is the average of the flips I actually made, which is .4. X1 is the value of the first flip, which in this case happens to be 1, but hopefully it's clear that E(X1) = .5 before I actually flip the coin since X1 is just an arbitrary flip of the coin.
 

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