189. Find the mathematical expectation of the random variable $Z$, if the mathematical expectations of $X$ and $Y$ are known:

a) $Z=X+2 Y, M(X)=5, M(Y)=3 ;$ b) $Z=3 X+4 Y$, $M(X)=2, M(Y)=6$
Solution. a) Using the properties of mathematical expectation (the expectation of a sum is equal to the sum of the expectations of the terms; a constant factor can be factored out of the expectation), we get

$$
\begin{aligned}
M(Z)=M(X+2 Y)= & M(X)+M(2 Y)=M(X)+2 M(Y)= \\
& =5+2 \cdot 3=11 .
\end{aligned}
$$