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, \(\displaystyle{E}^{{2}}{\left({70}-{2}{X}\right)}={70}^{{2}}+{4}{E}^{{2}}{\left({X}\right)}-{2}\times{70}\times{2}{E}{\left({X}\right)}\)

\(\displaystyle={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}^{{2}}{\left({X}\right)}\)

\(\displaystyle\Rightarrow{V}{\left({70}-{2}{X}\right)}={70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}{\left({X}^{{2}}\right)}-{\left({70}^{{2}}-{280}{E}{\left({X}\right)}+{4}{E}^{{2}}{\left({X}\right)}\right)}\)

\(\displaystyle={4}{\left[{E}{\left({X}^{{2}}\right)}-{E}^{{2}}{\left({X}\right)}\right]}\)

\(\displaystyle={4}\times{V}{\left({X}\right)}={4}\times{13}={52}\)