Question

If a variable takes values 0, 1, 2,., n with the frequencies 1, nc1,nc2,....,ncn, then their mean is

Answer 1

Answer:

n.2ⁿ⁻¹

Step-by-step explanation:

Hi,

Mean of a random variable is defined as μ = ∑xiP(x=xi).

X = xi        0          1         2         3 .....................n

P(X=xi)    nC0     nC1     nC2    nC3................nCn

μ = 0* nC0 + 1*nC1 +2*nC2+........+n*nCn

 = ∑i = 0 to n  i*nCi

Now, consider binomial expansion of

(1 + x)ⁿ = nC0 + x*nC1 +x²nC2 +........+nCnxⁿ

Differentiate the above equation w.r.t x on both sides,

we get

n(1+x)ⁿ⁻¹ = nC1 + 2xnC2 +............ + n*nCnxⁿ⁻¹,

Put x =1, we get

n.2ⁿ⁻¹ =  ∑i = 0 to n  i*nCi,

Thus , μ = n.2ⁿ⁻¹.

Hope, it helped !