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Question

If the variable takes values 0,1,2,3,...,n with frequencies, proportional to nC0, nC1, nC2,..., nCn respectively, then variance is

A
n4
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B
n3
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C
2n5
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D
None of these
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Solution

The correct option is A n4
xi=0,1,2,...,n
fi=nC0,nC1,nC2,...,nCn respectively.
Variance =x2ififi(xififi)2
Now we can solve it one by one
(xififi)=⎜ ⎜ ⎜ ⎜nr=0rnCrnr=0nCr⎟ ⎟ ⎟ ⎟=12n(nr=0rnCr)
=12n(nr=0rnrn1Cr1)=n2n(nr=0n1Cr1)
=n2n(2n1)=n2

Now,
(x2ififi)=⎜ ⎜ ⎜ ⎜nr=0r2(nCr)nr=0nCr⎟ ⎟ ⎟ ⎟=12n(nr=0r2(nCr))
=12n(nr=0(r2+rr)nCr)
=12n(nr=0(r(r1))nCr)+12n(nr=0(r)nCr)
=n24n4+n2

So, Variance =x2ififi(xififi)2
=n24n4+n2(n2)2=n4



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