i have a variables X and Y which are discrete, independant and uniformly distributed,$$X\sim[1,5],Y\sim[1,7]$$ I want to derive the variance of $$\frac{(X+Y)}2$$, but I don't know how, any help is much appreciated.
Not sure if it helps but $E[X]=3$ , $E[X^2]=\frac{91}{6}$ , $E[Y]=4$ , $E[Y^2]=20$
$$\mathsf {Var}(\tfrac{X+Y}{2}) = \mathsf E(\tfrac{(X+Y)^2}{4})-{\mathsf E(\tfrac {X+Y}2)}^2$$Well, you have $\mathsf E(X), \mathsf E(X^2), \mathsf E(Y), \mathsf E(Y^2)$, and by Linearity of Expecation:
$$\begin{split}\mathsf E(\tfrac{X+Y}2)&=\frac{\mathsf E(X)+\mathsf E(Y)}{2} \\[2ex] \mathsf E(\tfrac{(X+Y)^2}{4})&= \frac{\mathsf E(X^2)+2\mathsf E(XY)+\mathsf E(Y^2)}{4}\end{split}$$
This leaves you to find $\mathsf E(XY)$ when $X,Y$ are independent uniform distributions.