ISS PREVIOUS YEAR 2016 PAPER-1 SET-A Q.no.- 5

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

ISS Previous year question 2016

Question 5:-

For the distribution:

f(x) = [1 / B(p, q)] × [x^(p − 1) / (1 + x)^(p + q)],where 0 < x < ∞, p > 0, q > 0,

find the harmonic mean.

Options:

(a) p / (p + q) (b) 1 / p (c) (p − 1) / q (d) (p + 1) / (q − 1)

Solution

For a positive random variable X, the harmonic mean is defined as:

H = 1 / E(1/X)

So first we find E(1/X).

Step 1: Write expectation

E(1/X) = ∫ (1/x) f(x) dx from 0 to ∞

Substitute f(x):

E(1/X) = (1 / B(p, q)) × ∫ [1/x × x^(p−1) / (1 + x)^(p+q)] dx

Step 2: Simplify expression

1/x × x^(p−1) = x^(p−2)

So,

E(1/X) = (1 / B(p, q)) × ∫ [x^(p−2) / (1 + x)^(p+q)] dx

Step 3: Use Beta function identity

We know:

B(m, n) = ∫ [x^(m−1) / (1 + x)^(m+n)] dx from 0 to ∞

Compare with given integral:

x^(p−2) = x^(m−1) ⇒ m = p − 1m + n = p + q ⇒ n = q + 1

So,

∫ [x^(p−2) / (1 + x)^(p+q)] dx = B(p − 1, q + 1)

Step 4: Substitute back

E(1/X) = B(p − 1, q + 1) / B(p, q)

Step 5: Use Gamma function relation

B(a, b) = Γ(a)Γ(b) / Γ(a + b)

So,

E(1/X) = [Γ(p−1) Γ(q+1)] / [Γ(p) Γ(q)]

Step 6: Simplify using Gamma properties

Γ(q+1) = q Γ(q)Γ(p) = (p−1) Γ(p−1)

Substitute:

E(1/X) = [Γ(p−1) × q Γ(q)] / [(p−1) Γ(p−1) Γ(q)]

Cancel common terms:

E(1/X) = q / (p − 1)

Step 7: Find harmonic mean

H = 1 / E(1/X)

H = (p − 1) / q

Final Answer

Harmonic mean = (p − 1) / q

Correct option: (c)

Pro Tip

👉 This is a Beta Prime Distribution type question👉 Always remember:

E(1/X) = q / (p − 1)

Harmonic Mean = (p − 1) / q

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