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

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, $$\displaystyle{E}^{{2}}{\left({70}-{2}{X}\right)}={70}^{{2}}+{4}{E}^{{2}}{\left({X}\right)}-{2}\times{70}\times{2}{E}{\left({X}\right)}$$
$$\displaystyle={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}^{{2}}{\left({X}\right)}$$
$$\displaystyle\Rightarrow{V}{\left({70}-{2}{X}\right)}={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}{\left({X}^{{2}}\right)}-{\left({70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}^{{2}}{\left({X}\right)}\right)}$$
$$\displaystyle={4}{\left[{E}{\left({X}^{{2}}\right)}-{E}^{{2}}{\left({X}\right)}\right]}$$
$$\displaystyle={4}\times{V}{\left({X}\right)}={4}\times{13}={52}$$