208. Random variables $X$ and $Y$ are independent. Find the variance of the random variable $Z=3X+2Y$, given that $D(X)=5, D(Y)=6$.
Solution. Since the variables $X$ and $Y$ are independent, the variables $3X$ and $2Y$ are also independent. Using the properties of variance (the variance of the sum of independent random variables is equal to the sum of the variances of the addends; a constant factor can be factored out of the variance sign, squaring it), we get

$D(Z)=D(3X+2Y)=D(3X)+D(2Y)=9D(X)+4D(Y)=9 \cdot 5+4 \cdot 6=69$.