# Thread: Stuck at the therem

1. ## Stuck at the therem

Hi guys i am trying to solve the following theorem

Theorem : If $\displaystyle X_1,X_2,...X_n$ constitute the random sample from an infinite population with the mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$, then

$\displaystyle E[\bar X] = \mu$ and $\displaystyle var(\bar X) = \frac{\sigma^2}{n}$

Solution: Let $\displaystyle Y = \bar X$
$\displaystyle Y = \sum a_i \ X_i$

$\displaystyle \therefore$ $\displaystyle E[\bar X] = E \left( \sum a_i \ X_i \right)$

Let $\displaystyle a_i = \frac{1}{n}$ and as $\displaystyle E[\bar X] = \mu$ therefore substituting these terms in the above equation we get

$\displaystyle E[\bar X] = \sum_{n=1}^{\infty} \frac{1}{n} \mu$

Now after this in the book it shows this step

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n} \mu$ = $\displaystyle n. \frac{1}{n} \mu$...............How did he get 'n' instead of $\displaystyle \sum_{n=1}^{\infty}$

2. Because $\displaystyle \bar X={ \sum_{i=1}^n X_i \over n}$

The sample mean is the average of n, not an infinite number of observations.
and it's THEOREM

3. Originally Posted by matheagle
Because $\displaystyle \bar X={ \sum_{i=1}^n X_i \over n}$

The sample mean is the average of n, not an infinite number of observations.
and it's THEOREM
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n} \mu$ = n $\displaystyle . \frac{1}{n} \mu$
you didnt understand my question . I want to knw from where did the 'n' come from

4. Originally Posted by wolfyparadise
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n} \mu$ = n $\displaystyle . \frac{1}{n} \mu$
you didnt understand my question . I want to knw from where did the 'n' come from
Because you're summing n times a 'constant' (which doesn't depend on k)

5. $\displaystyle \sum_{n=1}^3k=k+k+k=3k$

$\displaystyle \sum_{n=1}^{100}k=100k$

$\displaystyle \sum_{n=1}^{100}\mu=100\mu$

6. Originally Posted by wolfyparadise
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n} \mu$ = n $\displaystyle . \frac{1}{n} \mu$
you didnt understand my question . I want to knw from where did the 'n' come from
I certainly DID understand your question.
YOU don't have the correct definition of the sample mean.

The sample mean is not an infinite sum

$\displaystyle \bar X={X_1+\cdots + X_n \over n}$

and

$\displaystyle E(\bar X)={E(X_1)+\cdots + (X_n) \over n}$

$\displaystyle ={\mu+\cdots + \mu \over n}={n\mu\over n}=\mu$

7. thanks