Find the value of $v a r (2 x_{1} - 3 x_{2})$ where $x_{1}$ and $x_{2}$ are two independent event with $E (x_{1}) = 2, E (x_{1}^{2}) = 7, E (x_{2}) = 3$ and $E (x_{2}^{2}) = 11$

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Solution

The correct option is C 30
$\begin{matrix} By property of variance: v a r (a x_{1} - b x_{2}) & = a^{2} v a r (x_{1}) + b^{2} v a r (x_{2}) - 2 a b c o v (x_{1} x_{2}) ∵ x_{1} a n d x_{2} are two independent event, then. c o v (x_{1} x_{2}) & = 0 So, v a r (2 x_{1} - 3 x_{2}) & = 4 v a r (x_{1}) + 9 v a r (x_{2}) v a r (x_{1}) & = E (x_{1}^{2} - [E (x_{1})]^{2} = 7 - 4 = 3 v a r (x_{2}) & = E (x_{2}^{2}) - [E (x_{2})]^{2} = 11 - 9 = 2 v a r (2 x_{1} - 3 x_{2} & = 4 \times 3 + 9 \times 2 = 30 \end{matrix}$

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