I was wondering if there is a shortcut or an easy way to calculate the expectation of a certain Random variable which is equal to two random variables with different probabilities.
For example: We are given random variables $X$ and $X$ (the numbers in the brackets are indexes) which have normal Gaussian distributions with $E(X)=E(X)=1$, $Var(X)=Var(X)=1$, and R.V $Y$ and $Y$ which have exponential distributions with parameter $lambda =2$, such that all of the R.V $X,X,Y$ and $Y$ are independent. If we define new random variables $Z$ and $Z$ such that $Z[i]$ equals $X[i]$ with probability $1/3$ and equals $Y[i]$ with probability $2/3$ for $i=1,2$, what is, for example, the expectation of $E(Z)$ and $E(Z\cdot Z)$?
I tried finding the probability distribution of both $Z\cdot Z$ and $Z$ and tried to calculate the respective expectations by integrating which is a rather long and gruesome procedure.
Thanks for any help in advance.