Hi i have the following question which may be a simple question for majority of the people. But i need to know the difference clearly ..
Let $X_1 ,...X_n$ be independently distributed (not identically) random variables with distribution of $ ~ uniform [0,n]$ .
I need to find the expectation of $ \sum_{k=1}^n X_k$
Since the distribution is $uniform$ , $E(X_k)= n/2$ . So $E(\sum_{k=1}^n X_k) = \sum_{k=1}^n E(X_k) $ = $\sum_{k=1}^n n/2$ .
Since the variables are only independently distributed, can i further simplify to $(n* n/2 )$ or do i need to stop from $\sum_{k=1}^n n/2$.
Also if the variables are $IID$ , then is $E(\sum_{k=1}^n X_k) = n* n/2 = n^2/2 $ ?
Thank you