If a variable x takes values $$0$$,$$1$$,$$2$$,…,n with frequencies proportional to the binomial coefficients $$nC0$$,$$nC1$$,$$nC2$$,… nCn , then var⁡(x) is

Question

If a variable x takes values $$0$$,$$1$$,$$2$$,…,n with frequencies  proportional to the binomial coefficients $$nC0$$,$$nC1$$,$$nC2$$,… nCn , then var⁡(x) is

Infinity Learn · Accepted Answer

We have x¯$$=0nC0+1nC1+2nC2+$$⋯$$+nnCn$$ $$nC0+nC1+nC2+$$⋯$$+nCn=$$∑$$r=0n$$ rnCr∑$$r=0n$$ nCr    $$=12n$$∑$$r=1n$$ rnrn−$$1Cr$$−$$1$$ ∵∑$$r=0n$$ $$nCr=2n$$;$$nCr=nrn$$−$$1Cr$$−$$1$$     $$=n2n$$∑$$r=1n$$ n−$$1Cr$$−$$1=n2n2n$$−$$1=n2$$ ∵∑$$r=1n$$ n−$$1Cr$$−$$1=2n$$−$$1$$ and $$1N$$∑$$fixi2=12n$$∑$$r=0nr2nCr=12n$$∑$$r=0n$$ [$$r(r$$−$$1)+r$$]$$nCr=12n$$∑$$r=0n$$ $$r(r$$−$$1)$$ $$Cr+$$∑$$r=0n$$ r $$Cr=12n$$∑$$r=2n$$ $$r(r$$−$$1)nrn$$−$$1r$$−$$1n$$−$$2Cr$$−$$2+$$∑$$r=1n$$ rnrn−$$1Cr$$−$$1=12nn(n$$−$$1)2n$$−$$2+n2n$$−$$1=n(n$$−$$1)4+n2$$∴Var⁡$$(X)=1N$$∑$$fixi2$$−x¯$$2=n(n$$−$$1)4+n2$$−$$n24=n4$$

$If a variable x takes values 0, 1, 2, \dots, n {with frequencies proportional to the binomial coefficients}^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots$ $^{n} C_{n}, then var (x) is$

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If a variable x takes values 0,1,2,…,n with frequencies proportional to the binomial coefficients nC0,nC1,nC2,… nCn , then var⁡(x) is

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$If a variable x takes values 0, 1, 2, \dots, n {with frequencies proportional to the binomial coefficients}^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots$ $^{n} C_{n}, then var (x) is$