We draws 10 from a finite population of 30, without replacement! 15 members out of 30 have the number 1, 10 members have the number 2 and 5 members have the number 3.
The Expected value is $$\mathbb E(X)= 1 \cdot \frac12+2 \cdot \frac13+3 \cdot \frac 16 = \frac 53 $$
Variance $$V(X)= \left( 1 -\frac53 \right)^2 \cdot \frac12+ \left( 2-\frac53 \right)^2 \cdot \frac13+ \left( 3-\frac53 \right)^2 \cdot \frac16 = \frac59 $$
I need the Variance of $$X_{1}+...+X_{10}$$ and $$Var( \frac1{10}( X_{1}+...+X_{10}))$$
I found this formula $$V(X_{1}+...+X_{10})= \sum\limits_{i=1}^{10} \operatorname{Var}(X_i)+\sum\limits_{i\neq j}^{} \operatorname{Cov}(X_i,X_j)$$
But how i can solve this? With Wolframalpha?
thx