Question

A coin is tossed three times. If Z is a random variable giving the number of heads obtained, (i) find the probability distribution of Z. (ii) what is the expected value of Z?

Answer 1

(i)

$$\def\arraystretch{1.5} \begin{array}{c:c:c} z &Outcomes & p(z) \\ \hline 0 & TTT & 1/8 \\ \hdashline 1 & HTT, THT, TTH & 3/8 \\ \hdashline 2 & HHT, HTH, THH & 3/8 \\ \hdashline 3 & HHH & 1/8 \\ \end{array}$$

$$\begin{matrix} z & 0 & 1 & 2 & 3 \\ p(z) & 1/8 & 3/8 & 3/8 & 1/8 \end{matrix}$$

(ii)

$$E[Z]=0(\dfrac{1}{8})+1(\dfrac{3}{8})+2(\dfrac{3}{8})+3(\dfrac{1}{8})=\dfrac{3}{2}=1.5$$