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  • If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

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Solution

Given-

Radom variable X is the member of tails in three tosses of a coin

Therefore, X= 0,1,2,3

\\ \Rightarrow \mathrm{P}(\mathrm{X}=\mathrm{x})=^\mathrm{n}C_{\mathrm{x}}(\mathrm{p})^{\mathrm{n}} \mathrm{q}^{\mathrm{n}-\mathrm{x}} \\ \text { Where } \mathrm{n}=3, \mathrm{p}=\frac{1}{2}, \mathrm{q}=\frac{1}{2} \text { and } \mathrm{x}=0,1,2,3

\begin{array}{|l|c|c|l|l|} \hline \mathrm{X} & 0 & 1 & 2 & 3 \\ \hline \mathrm{P}(\mathrm{X}) & \frac{1}{8} & \frac{3}{8} & \frac{3}{8} & \frac{1}{8} \\ \hline \mathrm{XP}(\mathrm{X}) & 0 & \frac{3}{8} & \frac{3}{4} & \frac{3}{8} \\ \hline \mathrm{X}^{2} \mathrm{P}(\mathrm{X}) & 0 & \frac{3}{8} & \frac{3}{2} & \frac{9}{8} \\ \hline \end{array}

\begin{aligned} &\text { As we know, Var }(X)=E\left(X^{2}\right)-[E(X)]^{2} \ldots \ldots \text { (i) }\\ &\text { Where, } E\left(X^{2}\right)=\sum_{i=1}^{n} x_{i}^{2} P(x) \text { and } E(X)=\sum_{i=1}^{n} x_{i} P\left(x_{i}\right)\\ &\therefore\left[\mathrm{E}\left(\mathrm{X}^{2}\right)\right]=\sum_{i=1}^{\mathrm{n}} \mathrm{x}_{i}^{2} \mathrm{P}\left(\mathrm{X}_{i}\right)\\ &=0+\frac{3}{8}+\frac{3}{2}+\frac{9}{8}=\frac{24}{8}\\ &=3\\ &\text { And }[\mathrm{E}(\mathrm{X})]^{2}=\left[\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{x}_{i} \mathrm{P}\left(\mathrm{X}_{\mathrm{i}}\right)\right]^{2} \end{aligned}

\\=\left[0+\frac{3}{8}+\frac{3}{4}+\frac{3}{8}\right]^{2} \\=\left[\frac{12}{8}\right]^{2}=\frac{144}{64} \\=\frac{9}{4}
Substituting the values of E(X)^{2}and [E(X)]^{2} in equation (i), we get:
\\\therefore \operatorname{Var}(\mathrm{X})=3-\frac{9}{4}=\frac{3}{4}
And standard deviation of \\ \mathrm{X}=\sqrt{\operatorname{Var}(\mathrm{X})} =\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}

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infoexpert22

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