# Independent random variables X_{1} and X_{2} are combined according to the formula L = 3*X_{1} + 2*X_{2}. If X_{1} and X_{2} both have a variance of 2.0, what is the variance of L?

Independent random variables ${X}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{X}_{2}$ are combined according to the formula $L=3\cdot {X}_{1}+2\cdot {X}_{2}$.
If ${X}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{X}_{2}$ both have a variance of 2.0, what is the variance of L?
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Usamah Prosser
Step 1
${X}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{X}_{2}$ are independent random variables.
$V\left({X}_{1}\right)=V\left({X}_{2}\right)=2$
$COV\left({X}_{1},{X}_{2}\right)=0,as{X}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{X}_{2}$ are independent random variables.
$V\left(L\right)=V\left(3{X}_{1}+2{X}_{2}\right)=9V\left({X}_{1}\right)+4V\left({X}_{2}\right)+12COV\left({X}_{1},{X}_{2}\right)$
$=9\cdot 2+3\cdot 2$
$=13\cdot 2=26$
$V\left(L\right)=26$