If p.m.f. of a r.v.X is $P(X=x)=\frac{1}{10}$, for $x=1,2,3, \ldots \ldots, 10=0$, otherwise, then…

If p.m.f. of a r.v.X is $P(X=x)=\frac{1}{10}$, for $x=1,2,3, \ldots \ldots, 10=0$, otherwise, then $\operatorname{Var}(x)$ is equal to
  1. $\frac{11}{2}$
  2. $\frac{33}{4}$
  3. $\frac{121}{4}$
  4. $\frac{77}{2}$

Solution

$\begin{aligned} & \operatorname{Var}(x)=\sum P_i x_i^2-\left(\sum P_i x_i\right)^2=\frac{1}{10}\left(1^2+2^2+3^2 \ldots .+10^2\right) \\ & -\left\{\frac{1}{10}(1+2+3+\ldots . .+10)\right\}^2 \\ & =\frac{1}{10} \times \frac{10 \times 11 \times 21}{6}-\left(\frac{1}{10} \times \frac{10 \times 11}{2}\right)^2 \\ & =38.5-30.25=8.25=\frac{33}{4}\end{aligned}$

Asked in: MHT CET 2022 (10 Aug Shift 1)

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