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

If a variable X takes values 0, 1, 2, ... , n  with frequencies proportional to the binomial coefficients 

 nC0,nC1,nC2,,nCn, then the  Var(X) is

  1. A

    n2112

  2. B

    n2

  3. C

    n4

  4. D

    none of these

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    Solution:

    We have,

    X¯=0×nC0×+1×nC1+2×nC2++n×nCn nC0+nC1+nC2++nCn X¯=r=0nr×nCrr=0nnCr X¯=12nr=1nr×nrn1Cr1                    r=0nnCr=2n; nCr=nrn1Cr1     X¯=n2nr=1nn1Cr1 X¯=n2n2n1=n2                                 r=1nn1Cr1=2n1

    and, 1Nfixi2=12nr=0nr2nCr

     

    1Nfixi2=12nr=0n{r(r1)+r}nCr 1Nfixi2=12nr=0nr(r1)nCr+r=0nrnCr 1Nfixi2=12nr=2nr(r1)nr×n1r1n2                                                                     +r=1nCr2rnn1rCr1

     1Nfixi2=12nn(n1)r=2nn2Cr2+nr=1nn1Cr1 1Nfixi2=12nn(n1)2n2+n2n1 =n(n1)4+n2 Var(X)=1Nfixi2X¯2=n(n1)4+n2n24=n4

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