Question

μ'_(r) and μ'_r represent the factorial moment of order r about the origin and r^th moment about the origin of the distribution x_i|f_i, i = 1, 2, … n. The value of μ'₂ equals to:
1. μ'₍₁₎²
2. μ'₍₂₎ - μ'₍₁₎
3. μ'₍₂₎ + μ'₍₁₎
4. μ'₍₂₎

Answer 1

Correct Answer - Option 3 : μ'₍₂₎ + μ'₍₁₎

Explanation

The moment about origin of binomial distribution is given by

 μ'₂ = E(X²) = ∑x²(n/x)p^xq^{n – x}

⇒ ∑{x(x – x) + x}[n(n – 1)/x(x – 1)](n – 2/x – 2)p^xq^{3 – x}

⇒ n(n – 1)p²[∑ⁿ_{x= 2}(n – 2/x – 2)p^{x – 2}q^{n – x]} + np

⇒ n(n -1)p² × ( q + p)^{n – 2} + np

⇒ n(n – 1)p² + np

⇒ n²p² – np² + np

Factorial moment of binomial distribution

μ’_(r) = E(x^(r)) = ∑ⁿ_{x = 0 ×} x²p(x)

⇒ ∑ⁿ_{x = 0} × x^(r)[n!/x!(n – x)!] × p^{x – r}_q^{n – r}

⇒ n^(r)p^r∑ⁿ_{x = r} [(n – r)!/(x – r)(n – x)!] × p^{x – r}q^{n – x}

⇒ n^(r)p^r(q+ p)^{n – r}

⇒ n^(r)p^(r)      [ p + q] = 1

⇒ μ’₍₁₎ = E(x¹)= np = mean

⇒ μ’₍₂₎ = E(x²) = n²p² =n(n – 1)p²

⇒ n²p² – np²

μ’₍₂₎ + μ’₍₁₎ = n²p² _– np² + np

∴ μ’₍₂₎+ μ’₍₁₎ = μ’₂