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Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]3 Marks
Question
Let the $p.m.f.$ of $r.v. X$ be $P(x)\left\{\begin{array}{l}\frac{3-x}{10}, \text { for } x=-1,0,1,2 \\ 0, \text { otherwise }\end{array}\right.$ Calculate $E(X)$ and Var$(X)$
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Answer

$ E ( X )=\sum_{ i =1}^4 x_{ i } P \left(x_{ i }\right)$
$=-1 \times\left(\frac{3-(-1)}{10}\right)+0 \times\left(\frac{3-0}{10}\right)+1 \times\left(\frac{3-1}{10}\right)+2 \times\left(\frac{3-2}{10}\right)$
$=\frac{-4+0+2+2}{10}$
$=0$
$E \left( X ^2\right)=\sum_{ i =1}^4 x_{ i }^2 P \left(x_{ i }\right)$
$=(-1)^2 \times \frac{4}{10}+(0)^2 \times \frac{3}{10}+(1)^2 \times \frac{2}{10}+(2)^2 \times \frac{1}{10}$
$=\frac{4+0+2+4}{10}$
$=1$
$\operatorname{Var}( X )= E \left( X ^2\right)-[ E ( X )]^2$
$=1-(0)^2$
$=1$

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