Question

If a variable takes values $0,1,2,3,....,n$ with frequencies $q^n,{}^n{C}_1{q}^{n-1}p,{}^n{C}_2{q}^{n-2}{p}^2,...,{}^n{C}_n{p}^n$ where $p+q=1$, then the mean is:

• A. $np$
• B. $nq$
• C. $n(p+q)$
• D. None of these

Answer 1

The correct option is A $np$

The required mean is
$\overline{x}=\frac{0.q^n+1.\frac{n}{1}.q^{n-1}.p+2.\frac{n(n-1)}{2!}.q^{n-2}.p^2+...+n.p^n}{q^n+\frac{n}{1}.q^{n-1}.p+\frac{n(n-1)}{2!}.q^{n-2}.p^2+...+p^n}$

$=\frac{{0.}^n{C}_0.q^n.p^0+{1.}^n{C}_1.q^{n-1}.p+...+n{.}^n{C}_n.q^0.p^n}{{}^n{C}_0.q^n.p^0{+}^n{C}_1.q^{n-1}.p+...{+}^n{C}_n.q^0.p^n}$

$=\frac{{\sum }_{r=0}^{n}r{.}^n{C}_r{q}^{n-r}{p}^r}{{\sum }_{r=0}^n{}^n{C}_r{q}^{n-r}{p}^r}=\frac{{\sum }_{r=0}^{n}r.\frac{n}{r}{.}^{n-1}{C}_{r-1}.q^{n-r}.p.p^{r-1}}{{\sum }_{r=0}^n{}^n{C}_r.q^{n-r}.p^r}$

$=\frac{np\left({\sum }_{r=1}^n{}^{n-1}{C}_{r-1}.p^{r-1}.q^{(n-1)-(r-1)}\right)}{{\sum }_{r=0}^n{}^n{C}_r.q^{n-r}.p^r}=\frac{np{(p+q)}^{n-1}}{{(p+q)}^n}=np$