ISS PREVIOUS YEAR 2016 PAPER-1 SET-A Q.no.- 6

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

ISS Previous year question 2016

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

Question 6 :-

Poisson Distribution Question with Solution (ISS Level)

A Poisson random variable X has mean equal to 1/2.

Let Y = 2X. Consider the following:

1. E(Y) = 1

2. Var(Y) = 4

3. μ3(Y) = 4

4. μ4(Y) = 28

Which of the above is/are correct?

Options:

(a) 1 only(b) 2 and 4 only(c) 1 and 3 only(d) 1, 2 and 3 only

Solution

Given:

X follows Poisson distribution with parameter λ = 1/2

So, we know:

• Mean: E(X) = λ = 1/2

• Variance: Var(X) = λ = 1/2

• Third central moment: μ3(X) = λ = 1/2

• Fourth central moment: μ4(X) = λ + 3λ²

Step 1: Mean of Y = 2X

E(Y) = E(2X)= 2E(X)= 2 × (1/2)= 1

✔ Statement 1 is correct

Step 2: Variance of Y

Var(Y) = Var(2X)= (2²) Var(X)= 4 × (1/2)= 2

Given value is 4 → Incorrect

Step 3: Third Central Moment

μ3(Y) = (2³) μ3(X)= 8 × (1/2)= 4

✔ Statement 3 is correct

Step 4: Fourth Central Moment

First calculate μ4(X):

μ4(X) = λ + 3λ²= (1/2) + 3 × (1/2)²= (1/2) + 3/4= 5/4

Now,

μ4(Y) = (2⁴) μ4(X)= 16 × (5/4)= 20

Given value is 28 → Incorrect

Final Answer

Statements 1 and 3 are correct

Correct option: (c) 1 and 3 only

Important Concept (Very Useful for ISS Exam)

If Y = aX, then:

• E(Y) = aE(X)

• Var(Y) = a²Var(X)

• μr(Y) = a^r μr(X)

This concept is frequently asked in ISS examination.

ISS PREVIOUS YEAR 2016 PAPER-1 SET-A

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