Fill in the blanks. Let X be a random variable taking values x_1,… - Vidyadip

Question

Fill in the blanks.
Let X be a random variable taking values $x_1, x_2, ......, x_n$ with probabilities $p_1, p_2, ..., p_n,$ respectively. Then var (X) = ________.

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Answer

Let X be a random variable taking values $x_1, x_2, ...,x_n$ with probabilities $p_1, p_2, ..., p_n$, respectively.
Then $\text{Var}(\text{X})=\sum\text{P}_{\text{i}}\text{x}_{\text{i}}^2-\Big(\sum\text{P}_{\text{i}}\text{x}_{\text{i}}\Big)^2.$
Solution:
$\text{Var}(\text{X})=(\text{X}^2)-\big[\text{E}(\text{X})\big]^2$
$=\sum_\limits{\text{i}=1}^\text{n}\text{X}^{2}\text{P}(\text{X})-\bigg[\sum_\limits{\text{i}=1}^\text{n}\text{XP}(\text{X})\bigg]^2$
$=\sum\text{P}_{\text{i}}\text{x}_{\text{i}}^2-\Big(\sum\text{P}_{\text{i}}\text{x}_{\text{i}}\Big)^2$

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