Question

Suppose X is a random variable such that E(3X-7)=8 and E(frac(X^2)(2))=19 .What is Var(70-2X)?

Multivariable functions
ANSWERED
asked 2021-09-15

Suppose X is a random variable such that \(E(3X-7)=8\) and \(\displaystyle{E}{\left({\frac{{{X}^{{2}}}}{{{2}}}}\right)}={19}\).
What is Var\((70-2X)\)?

Expert Answers (1)

2021-09-16

Step 1
We have,
\(E(3X-7)=8\)
\(3E(X)-7=8\)
\(3E(X)=15\)
\(\displaystyle{E}{\left({X}\right)}={\frac{{{15}}}{{{3}}}}={5}\)
Also,
\(\displaystyle{E}{\left({\frac{{{X}^{{2}}}}{{{2}}}}\right)}={19}\)
\(\displaystyle{E}{\left({X}^{{2}}\right)}={19}\times{2}={38}\)
Then , \(\displaystyle{V}{\left({X}\right)}={E}{\left({X}^{{2}}\right)}-{E}^{{2}}{\left({X}\right)}={38}-{5}^{{2}}={38}-{25}={13}\)
Step 2
\(\displaystyle{V}{\left({70}-{2}{X}\right)}={E}{\left({70}-{2}{X}\right)}^{{2}}-{E}^{{2}}{\left({70}-{2}{X}\right)}\)
\(\displaystyle{E}{\left({70}-{2}{X}\right)}^{{2}}={E}{\left[{70}^{{2}}-{2}\times{70}\times{2}{X}+{4}{X}^{{2}}\right]}\)
\(\displaystyle={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}{\left({X}^{{2}}\right)}\)
\(\displaystyle{E}{\left({70}-{2}{X}\right)}={70}-{2}{E}{\left({X}\right)}\)
Then, \(\displaystyle{E}^{{2}}{\left({70}-{2}{X}\right)}={70}^{{2}}+{4}{E}^{{2}}{\left({X}\right)}-{2}\times{70}\times{2}{E}{\left({X}\right)}\)
\(\displaystyle={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}^{{2}}{\left({X}\right)}\)
\(\displaystyle\Rightarrow{V}{\left({70}-{2}{X}\right)}={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}{\left({X}^{{2}}\right)}-{\left({70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}^{{2}}{\left({X}\right)}\right)}\)
\(\displaystyle={4}{\left[{E}{\left({X}^{{2}}\right)}-{E}^{{2}}{\left({X}\right)}\right]}\)
\(\displaystyle={4}\times{V}{\left({X}\right)}={4}\times{13}={52}\)

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