# Expected values of random variables !

#### sid9221

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

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

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#### Office_Shredder

Staff Emeritus
Gold Member
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

Staff Emeritus
Gold Member
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

Staff Emeritus
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:
$\mu$ 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.