Question

The standard deviation of the data:

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

is

Options

• \[\left( \frac{1 + a^2}{2} \right)^n - \left( \frac{1 + a}{2} \right)^n\]

   

•  \[\left( \frac{1 + a^2}{2} \right)^{2n} - \left( \frac{1 + a}{2} \right)^n\]

•  \[\left( \frac{1 + a}{2} \right)^{2n} - \left( \frac{1 + a^2}{2} \right)^n\]

   

• none of these

   

Answer 1

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 \[^{n}{}{C}_n\]
	\[\sum^n_{i = 1} f_i = 2^n\]	\[\sum^n_{i = 1} f_i x_i = \left( 1 + a \right)^n\]		\[\sum^n_{i = 1} f_i {x_i}^2 = \left( 1 + a^2 \right) {}^n\]

`"Number of terms," N = \sum_{i = 1}^2  f_i = 2^n `

` \sum _{i = 1}^2 f_i x_i = ^nC_0 + a ^nC_1 + a^2 "^nC_2 + . . . + a"^n "^nC_n = \left( 1 + a \right)^n `

`X = \frac{\sum_{i = 1}^n f_ix_i}{N}`

\[ = \frac{\left( 1 + a \right)^n}{2^n}\]

` \sum_{i = 1}^n f_i x_i^2 = \left( 1 + a^2 \right)^n`

`\sigma^2 = \text{ Variance } \left( X \right) = \frac{1}{N} \sum_{i = 1}^n f_i_x_i^2 - \left( {\sum_{i = 1}^n f_i x_i}/{N} \right)^2 `

\[ = \frac{\left( 1 + a^2 \right)^n}{2^n} - \left[ \frac{\left( 1 + a \right)^n}{2^n} \right]^2 \]

\[ = \left[ \frac{1 + a^2}{2} \right]^n - \left[ \frac{1 + a}{2} \right]^{2n} \]

\[\sigma = \sqrt{\text{ Variance }  \left( X \right)} \]

\[ = \sqrt[]{\left[ \frac{1 + a^2}{2} \right]^n - \left[ \frac{1 + a}{2} \right]^{2n}}\]