# Thread: mean and variance of population distribution - law of large numbers - plz help

1. ## mean and variance of population distribution - law of large numbers - plz help

A population distribution has a mean of 10 and a variance of 16. Determine the mean and variance of Ybar from an i.i.d. sample from this population for a) n=10, b) n= 100 and c) n=1000. Relate the answers to the law of large numbers.

2. ## Re: mean and variance of population distribution - law of large numbers - plz help

Originally Posted by homalina
A population distribution has a mean of 10 and a variance of 16. Determine the mean and variance of Ybar from an i.i.d. sample from this population for a) n=10, b) n= 100 and c) n=1000. Relate the answers to the law of large numbers.

For iid $Y_1,\cdots, Y_n$ with mean and variance $\mu\;&\;\sigma^2$
$\bar{Y}=\dfrac{1}{n}\sum_{i=1}^n Y_i=\mu$

$E[\bar{Y}]=\dfrac{1}{n}\sum_{i=1}^n E[Y_i]=\mu$

also, $Var(\bar{Y})=\dfrac{1}{n^2}\sum_i {\sigma_i}^2=...$

Thanks a lot