Question

If a coin is tossed three times and $X=$ number of heads, then find $E\left(X\right)$ and $V\left(X\right)$.

Answer 1

$S=\{HHH,HHT,HTT,HTH,TTT,TTH,THT,THH\}$

coin is tossed three time

$2^3=8,x=0,1,2,3$

$P(x=0)=P\left(TTT\right)=\frac{1}{8}$

$P(x=1)=P(x=1)=\frac{3}{8}$

$P(x=2)=\frac{3}{8}$

$P(x=3)=\frac{1}{8}$

$E\left(x\right)=\Sigma x_i{f}_i$

$=0\times \frac{1}{8}+1\times \frac{3}{8}+2\times \frac{3}{8}+3\times \frac{1}{8}$

$=\frac{3+6+3}{8}$

$\therefore E\left(x\right)=\frac{12}{8}=\frac{3}{2}$

$V\left(x\right)=\Sigma {x_i}^2{f}_i$

$=0^2\times \frac{1}{8}+1^2\times \frac{3}{8}{2}^2\times \frac{3}{8}+3^2\times \frac{1}{8}$

$=\frac{3}{8}+\frac{12}{8}+\frac{9}{8}$

$=\frac{24}{8}$

$V\left(x\right)=3$