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−$$1Cr$$−$$1=2n$$−$$1and$$ $$1N$$∑$$fixi2=12n$$∑$$r2nCr=12n$$∑$$r=0n$$ [$$r(r$$−$$1)+r$$]nCr     $$=12n$$∑$$r=0n$$ r−$$(r$$−$$1)nCr+$$∑$$r=0n$$ rnCr     $$=12n$$∑$$r=2n$$ $$r(r$$−$$1)nrn$$−$$1r$$−$$1n$$−$$2Cr$$−$$2+$$∑$$r=1n$$ 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, . . ., 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, . . ., 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