Question

The standard deviation of the data:

x:	1	a	a2	....	an
f:	nC0	nC1	nC2	....	nCn

is
(a) ${\left(\frac{1+a^2}{2}\right)}^n-{\left(\frac{1+a}{2}\right)}^n$

(b) ${\left(\frac{1+a^2}{2}\right)}^{2n}-{\left(\frac{1+a}{2}\right)}^n$

(c) ${\left(\frac{1+a}{2}\right)}^{2n}-{\left(\frac{1+a^2}{2}\right)}^n$

(d) none of these

Answer 1

(d) None of these

xi	fi	fixi	${x_i}^2$	$f_i{x_i}^2$
1	${}^{n}C_0$	${}^{n}C_0$	1	1
a	${}^{n}C_1$	a ${}^{n}C_1$	a2	a2 ${}^{n}C_1$
a2	${}^{n}C_2$	a2 ${}^{n}C_2$	a4	a4 ${}^{n}C_2$
a3	${}^{n}C_3$	a3 ${}^{n}C_3$	a6	a6 ${}^{n}C_3$
: : : :	: : :	: : :	: : :	: : : :
an	${}^{n}C_n$	an ${}^{n}C_n$	a2n	a2n ${}^{n}C_n$
	$\sum _{i=1}^n{f}_i=2^n$	$\sum _{i=1}^n{f}_i{x}_i={\left(1+a\right)}^n$		$\sum _{i=1}^n{f}_i{x_i}^2=\left(1+a^2\right){}^n$

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{terms},N=\underset{}{\sum _{i=1}^n}{f}_i=2^n \\ \underset{}{\sum _{i=1}^n{f}_i}{x}_i={}^{n}C_0+a{}^{n}C_1+a^2{}^{n}C_2+...+a^n{}^{{}_n}C_n={\left(1+a\right)}^n \\ \overline{X}=\frac{\underset{}{{\displaystyle \sum _{i=1}^n}{f}_i}{x}_i}{N} \\ =\frac{{\left(1+a\right)}^n}{2^n} \\ \underset{}{\sum _{i=1}^n{f}_i}{x_i}^2={\left(1+a^2\right)}^n \\ {\sigma }^2=\mathrm{Variance}\left(\mathrm{X}\right)=\underset{}{\frac{1}{N}\sum _{i=1}^n{f}_i}{x_i}^2-{\left(\frac{\underset{}{{\displaystyle \sum _{i=1}^n}{f}_i}{x}_i}{N}\right)}^2 \\ =\frac{{\left(1+a^2\right)}^n}{2^n}-{\left[\frac{{\left(1+a\right)}^n}{2^n}\right]}^2 \\ ={\left[\frac{1+a^2}{2}\right]}^n-{\left[\frac{1+a}{2}\right]}^{2n} \\ \sigma =\sqrt{\mathrm{Variance}\left(X\right)} \\ =\sqrt[]{{\left[\frac{1+a^2}{2}\right]}^n-{\left[\frac{1+a}{2}\right]}^{2n}} \end{gathered}$$