Question

When three coins are tossed,the probability distribution for the random variable x representing the number of heads that occurs is given below compute the variance and standard deviation of the probability distribution

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} & x \\ \hline TTT & 0 \\ \hdashline TTH & 1\\ \hdashline THT & 1 \\ \hdashline HTT & 1 \\ \hdashline THH & 2 \\ \hdashline HTH & 2\\ \hdashline HHT & 2\\ \hdashline HHH & 3 \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 1/8 & 3/8 & 3/8 & 1/8 \\ \end{array}$$

$$E(X)=\dfrac{1}{8}(0)+\dfrac{3}{8}(1)+\dfrac{3}{8}(2)+\dfrac{1}{8}(3)=\dfrac{3}{2}$$

$$E(X^2)=\dfrac{1}{8}(0)^2+\dfrac{3}{8}(1)^2+\dfrac{3}{8}(2)^2+\dfrac{1}{8}(3)^2=3$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2$$

$$=3-(\dfrac{3}{2})^2=\dfrac{3}{4}$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{3}{4}}=\dfrac{\sqrt{3}}{2}\approx0.8660$$