(a) Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

Let $\displaystyle X$ denote the random variable which counts the number of boys. $\displaystyle X=0,1,2,3$ \[P(\text { Boy })=P(\text { Girl })=\frac{1}{2} \] Required Probability Distribution $\displaystyle P(X)$ & $\displaystyle \left(\frac{1}{2}\right)^{3}=\frac{1}{8}$ & $\displaystyle 3\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)^{2}=\frac{3}{8}$ & $\displaystyle 3\left(\frac{1}{2}\right)^{2}\left(\frac{1}{2}\right)=\frac{3}{8}$ & $\displaystyle \left(\frac{1}{2}\right)^{3}=\frac{1}{8}$ Possible values of $\displaystyle X$ are - $\displaystyle 2,0,2$Mean $\displaystyle =\sum X P(X)=-2\left(\frac{1}{4}\right)+0\left(\frac{1}{2}\right)+2\left(\frac{1}{4}\right)=0$