The variance of the data:
x:
$1$
$a$
$a^{2}$
......
$a^{n}$
f:
${^{n} C}_{0}$
${^{n} C}_{1}$
${^{n} C}_{2}$
......
${^{n} C}_{n}$
is

{(\frac{1 + a^{2}}{2})}^{n} - {(\frac{1 + a}{2})}^{n}

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{(\frac{1 + a^{2}}{2})}^{n} - {(\frac{1 + a}{2})}^{2 n}

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{(\frac{1 + a}{2})}^{2 n} - {(\frac{1 + a^{2}}{2})}^{n}

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none of these

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Solution

The correct option is B ${(\frac{1 + a^{2}}{2})}^{n} - {(\frac{1 + a}{2})}^{2 n}$
$¯ X = \frac{\sum f_{i} x_{i}}{\sum f_{i}} = \frac{^{n} C_{0} + a .^{n} C_{1} + a^{2} .^{n} C_{2} + . . . . a^{n} .^{n} C_{n}}{^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + . . . .^{n} C_{n}} = \frac{{(1 + a)}^{n}}{2^{n}}$
.
$S . D . = \sqrt{\frac{\sum f_{i} {x_{i}}^{2}}{\sum f_{i}} - {(\frac{\sum f_{i} x_{i}}{\sum f_{i}})}^{2}} = \sqrt{\frac{^{n} C_{0} + a^{2} .^{n} C_{1} + a^{4} .^{n} C_{2} + . . . . a^{2 n} .^{n} C_{n}}{^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + . . . .^{n} C_{n}} - {[\frac{{(1 + a)}^{n}}{2^{n}}]}^{2}} \sqrt{\frac{{(1 + a^{2})}^{n}}{2^{n}} - {(\frac{1 + a}{2})}^{2 n}}$

$V a r i a n c e = (S . D)^{2}$

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Similar questions

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

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x:	$1$	$a$	$a^{2}$	......	$a^{n}$
f:	${^{n} C}_{0}$	${^{n} C}_{1}$	${^{n} C}_{2}$	......	${^{n} C}_{n}$