Theoretical DistributionsMTP June 2023 Series IIQuestion 3975 of 230
All Questions

If Poisson distribution is such that P(X=2)=P(X=3)\displaystyle P(X=2) = P(X=3) then the Standard Deviation of the distribution is

Options

A3\displaystyle \sqrt{3}
B3\displaystyle 3
C6\displaystyle 6
D9\displaystyle 9
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Correct Answer

✅ Option a — 3\displaystyle \sqrt{3}

All Options:

  • A3\displaystyle \sqrt{3}
  • B3\displaystyle 3
  • C6\displaystyle 6
  • D9\displaystyle 9

Detailed Solution & Explanation

**Standard Deviation of Poisson Distribution given P(X=2)=P(X=3)\displaystyle P(X=2) = P(X=3)** Let m\displaystyle m be the parameter (mean and variance) of the Poisson distribution. The probability mass function is: P(X=x)=frace−mmxx!P(X = x) = \\frac{e^{-m} m^x}{x!} **Step 1: Write the expressions for P(X=2)\displaystyle P(X=2) and P(X=3)\displaystyle P(X=3)** - For x=2\displaystyle x = 2: P(X=2)=frace−mm22!=frace−mm22P(X = 2) = \\frac{e^{-m} m^2}{2!} = \\frac{e^{-m} m^2}{2} - For x=3\displaystyle x = 3: P(X=3)=frace−mm33!=frace−mm36P(X = 3) = \\frac{e^{-m} m^3}{3!} = \\frac{e^{-m} m^3}{6} **Step 2: Equate the two probabilities** According to the problem: P(X=2)=P(X=3)P(X = 2) = P(X = 3) frace−mm22=frace−mm36\\frac{e^{-m} m^2}{2} = \\frac{e^{-m} m^3}{6} **Step 3: Solve for m\displaystyle m** Since e−m>0\displaystyle e^{-m} > 0 and m>0\displaystyle m > 0, we can divide both sides by frace−mm22\displaystyle \\frac{e^{-m} m^2}{2}: 1=fracm3impliesm=31 = \\frac{m}{3} \\implies m = 3 **Step 4: Find the Standard Deviation (S.D.)** For a Poisson distribution, the variance is equal to its parameter m\displaystyle m: textVariance=m=3\\text{Variance} = m = 3 textStandardDeviation=sqrttextVariance=sqrt3\\text{Standard Deviation} = \\sqrt{\\text{Variance}} = \\sqrt{3} This corresponds to Option A. Hence, **Option A** 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.

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