ISS PREVIOUS YEAR 2016 PAPER-1 SET-A Q.no.- 8

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

ISS Previous year question 2016

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

Probability Question Solution

Question 8

Let XXX be a Poisson variate with parameter λ\lambdaλ such that

P(X = 2) = 2P(X = 4) + 20P(X = 6)

What is the coefficient of skewness?

Options:

(a) 1/√3(b) 1(c) 1/2(d) −1/√3

Solution

For a Poisson distribution with parameter λ\lambdaλ, the probability mass function is

P(X = k) = (e^(−λ) × λ^k) / k!

Step 1: Write the probabilities

P(X = 2) = (e^(−λ) × λ²) / 2!

P(X = 4) = (e^(−λ) × λ⁴) / 4!

P(X = 6) = (e^(−λ) × λ⁶) / 6!

Step 2: Substitute into the given equation

(e^(−λ) × λ²)/2! = 2[(e^(−λ) × λ⁴)/4!] + 20[(e^(−λ) × λ⁶)/6!]

Cancel e^(−λ) from both sides:

λ² / 2 = 2(λ⁴ / 24) + 20(λ⁶ / 720)

Step 3: Simplify

λ² / 2 = λ⁴ / 12 + λ⁶ / 36

Multiply both sides by 36:

18λ² = 3λ⁴ + λ⁶

Step 4: Rearranging

λ⁶ + 3λ⁴ − 18λ² = 0

Factor out λ²:

λ²(λ⁴ + 3λ² − 18) = 0

Let y = λ²

y² + 3y − 18 = 0

(y + 6)(y − 3) = 0

y = 3

Thus

λ² = 3λ = √3

Step 5: Coefficient of Skewness

For a Poisson distribution,

Skewness = 1 / √λ

Substitute λ = √3

Skewness = 1 / √3

Final Answer

Correct option: (a) 1/√3

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

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