Expected values of random variables

[sid9221]

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

 

[Office_Shredder]

X₁ is a random variable whose mean is $\mu$ by definition. Can you elaborate on your confusion?

 

[sid9221]

X₁ is a random variable whose mean is $\mu$ by definition. Can you elaborate on your confusion?

Where is this defined ? Is is part of the definition of 'Expectation' ?

 

[Office_Shredder]

Why don't you tell us what you think X₁ is, and what $\mu$ is, and we can work from there.

 

[sid9221]

Why don't you tell us what you think X₁ is, and what $\mu$ is, and we can work from there.

μ=$\frac{\sum X_i}{N}$

$x_1$ is just a variable

 

[Office_Shredder]

μ=$\frac{\sum X_i}{N}$

No, the thing on the right hand side is $\overline{X}$, not $\mu$. 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.

$\mu$ in this context is the expected value of a single flip of the coin, which is .5. $\overline{X}$ 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.