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

Given that, random variable X is the number of tails in three tosses of a coin.

So, $x = 0,1,2,3$

$\Rightarrow$ $P(X = x){ = ^n}{C_x}{(p)^x}{q^{n - x}}$,

where $n = 3,p = 1/2,q = 1/2$

and $x = 0,1,2,3$

We know that, ${\mathop{\rm Var}\nolimits} (X) = E\left( {{X^2}} \right) - {[E(X)]^2}$ …….(i)

where, $E\left( {{X^2}} \right) = \sum\limits_{i = 1}^n {x_i^2} P\left( {{x_i}} \right)$

and $E(X) = \sum\limits_{i = 1}^n {{x_i}} P\left( {{x_i}} \right)$

$\therefore$ $E\left( {{X^2}} \right) = \sum\limits_{i = 1}^n {x_i^2} P\left( {{X_i}} \right) = 0 + \frac{3}{8} + \frac{3}{2} + \frac{9}{8} = \frac{{24}}{8} = 3$

and ${[E(X)]^2} = {\left[ {\sum\limits_{i = 0}^n {x_i^2} P\left( {{x_i}} \right)} \right]^2} = {\left[ {0 + \frac{3}{8} + \frac{3}{4} + \frac{3}{8}} \right]^2} = {\left[ {\frac{{12}}{8}} \right]^2} = \frac{9}{4}$

$\therefore$ ${\mathop{\rm Var}\nolimits} (X) = 3 - \frac{9}{4} = \frac{3}{4}$ [using Eq. (i)]

and standard deviation of $X = \sqrt {{\mathop{\rm Var}\nolimits} (X)} = \sqrt {\frac{3}{4}} = \frac{{\sqrt 3 }}{2}$