Thread: Calculating variance of u_n - u_2n

Let u_n = (sum_(k=1,n) X_k)/n where X_i is an numerical observation of a random variable with density function f.

Similarily let u_2n = (sum_(k=1,2n) X_k)/2n where again X_i is an observation of a random variable with same density function f

Determine the value of E( (u_n - u_2n)^2 ) where 'E' represents the expected value.

Hello,

Let's assume that and are independent . If it is not, tell me and throw my solution in the bin

This implies that (it will be used later)






Therefore




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Properties of the variance :

, where a is a constant.


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Since the covariance between any and is 0, we can write :










I hope I have not mistaken somewhere...

Moo

Yes there is a mistake somewhere but its my fault for not clarifying in my initial thread.

Every single one of the X_i in the second sum (from 1 to 2n ) is different from those in the first sum of X_i (ie its not just the second half from n + 1 to 2n which are different).

I hope this helps.