If

nitraiddQ
2020-11-22
Answered

Independent random variables $X}_{1}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{X}_{2$ are combined according to the formula $L=3\cdot {X}_{1}+2\cdot {X}_{2}$ .

If$X}_{1}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{X}_{2$ both have a variance of 2.0, what is the variance of L?

If

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Usamah Prosser

Answered 2020-11-23
Author has **86** answers

Step 1

$X}_{1}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{X}_{2$ are independent random variables.

$V\left({X}_{1}\right)=V\left({X}_{2}\right)=2$

$COV({X}_{1},{X}_{2})=0,as{X}_{1}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{X}_{2}$ are independent random variables.

$V\left(L\right)=V(3{X}_{1}+2{X}_{2})=9V\left({X}_{1}\right)+4V\left({X}_{2}\right)+12COV({X}_{1},{X}_{2})$

$=9\cdot 2+3\cdot 2$

$=13\cdot 2=26$

$V\left(L\right)=26$

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