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?

Question
Random variables
asked 2020-11-22
Independent random variables \(\displaystyle{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}\) are combined according to the formula \(\displaystyle{L}={3}\cdot{X}_{{{1}}}+{2}\cdot{X}_{{{2}}}\).
If \(\displaystyle{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}\) both have a variance of 2.0, what is the variance of L?

Answers (1)

2020-11-23
Step 1
\(\displaystyle{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}\) are independent random variables.
\(\displaystyle{V}{\left({X}_{{{1}}}\right)}={V}{\left({X}_{{{2}}}\right)}={2}\)
\(\displaystyle{C}{O}{V}{\left({X}_{{{1}}},{X}_{{{2}}}\right)}={0},{a}{s}{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}\) are independent random variables.
\(\displaystyle{V}{\left({L}\right)}={V}{\left({3}{X}_{{{1}}}+{2}{X}_{{{2}}}\right)}={9}{V}{\left({X}_{{{1}}}\right)}+{4}{V}{\left({X}_{{{2}}}\right)}+{12}{C}{O}{V}{\left({X}_{{{1}}},{X}_{{{2}}}\right)}\)
\(\displaystyle={9}\cdot{2}+{3}\cdot{2}\)
\(\displaystyle={13}\cdot{2}={26}\)
\(\displaystyle{V}{\left({L}\right)}={26}\)
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