Theoretical DistributionsMCQPYQ Dec 22Question 3499 of 230
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If a Poisson distribution is such that P(X=2)=P(X=3)\displaystyle P(X=2) = P(X=3) then the variance of the distribution

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — 3

All Options:

  • A
    3\displaystyle \sqrt{3}
  • B3
  • C6
  • D9

Detailed Solution & Explanation

Let the Poisson parameter be m\displaystyle m. The probability mass function is: P(X=k)=e−mmkk!P(X = k) = e−mmkk!\displaystyle \frac{e^{-m} m^k}{k!} Given that: P(X=2)=P(X=3)P(X = 2) = P(X = 3) Substituting the formulas: e−mm22!=e−mm33!\displaystyle \frac{e^{-m} m^2}{2!} = \frac{e^{-m} m^3}{3!} e−mm22=e−mm36\displaystyle \frac{e^{-m} m^2}{2} = \frac{e^{-m} m^3}{6} Since e−m≠0\displaystyle e^{-m} ≠\displaystyle \neq 0 and m>0\displaystyle m > 0: 12=m6  ⟹  m=312=m6\displaystyle \frac{1}{2} = \frac{m}{6} \implies m = 3 Since the variance of a Poisson distribution is equal to its parameter m\displaystyle m, the variance is 3\displaystyle 3. Hence, **Option B** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Variance

A measure of dispersion equal to the average of the squared deviations of each data point from the mean. Formula: σ² = Σ(xᵢ − x̄)² / n. Standard Deviation = √Variance.

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