If a Poisson distribution is such that P(X=2) = P(X=3) , then the standard deviation of the distribution is:

Option a: sqrt(3). Solution: Let the Poisson parameter be m . The probability mass function is: P(X = k) = -m mkk! Given the condition: P(X = 2) = P(X = 3) Substituting the values: -m m22! = -m m33! m2/2 = m3/6 Since m > 0 : 1/2 = m/6 m = 3 The variance of the Poisson distribution is m = 3 . The standard deviation is sqrt(m) = sqrt(3) . Note: The source option key indicates Option B (which corresponds to the variance 3 ), but the question specifically asks for the standard deviation. Mathematically, the standard deviation is sqrt(3) , which corresponds to Option A. Hence, Option A is the correct answer.

If a Poisson distribution is such that $P(X=2) = P(X=3)$, then the standard deviation of the distribution is:

The correct answer is Option a: $\sqrt{3}$. Let the Poisson parameter be $m$. The probability mass function is: $$P(X = k) = \frac{e^{-m} m^k}{k!}$$ Given the condition: $$P(X = 2) = P(X = 3)$$ Substituting the values: $$\frac{e^{-m} m^2}{2!} = \frac{e^{-m} m^3}{3!} \implies \frac{m^2}{2} = \frac{m^3}{6}$$ Since $m > 0$: $$\frac{1}{2} = \frac{m}{6} \implies m = 3$$ The variance of the Poisson distribution is $m = 3$. The standard deviation is $\sqrt{m} = \sqrt{3}$. Note: The source option key indicates Option B (which corresponds to the variance $3$), but the question specifically asks for the standard deviation. Mathematically, the standard deviation is $\sqrt{3}$, which corresponds to Option A. Hence, **Option A** is the correct answer.

Theoretical DistributionsMCQPYQ Jun 23Question 3501 of 230

All Questions

If a Poisson distribution is such that $\displaystyle P(X=2) = P(X=3)$ , then the standard deviation of the distribution is:

Options

\displaystyle \sqrt{3}

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Correct Answer

✅ Option a — $\displaystyle \sqrt{3}$

All Options:

A $\displaystyle \sqrt{3}$
B3
C2
D1

Detailed Solution & Explanation

Let the Poisson parameter be

\displaystyle m

. The probability mass function is:

P(X = k) = \frac{e^{-m} m^k}{k!}

Given the condition:

P(X = 2) = P(X = 3)

Substituting the values:

\frac{e^{-m} m^2}{2!} = \frac{e^{-m} m^3}{3!} \implies \frac{m^2}{2} = \frac{m^3}{6}

Since

\displaystyle m > 0

\frac{1}{2} = \frac{m}{6} \implies m = 3

The variance of the Poisson distribution is

\displaystyle m = 3

. The standard deviation is

\displaystyle \sqrt{m} = \sqrt{3}

. Note: The source option key indicates Option B (which corresponds to the variance

\displaystyle 3

), but the question specifically asks for the standard deviation. Mathematically, the standard deviation is

\displaystyle \sqrt{3}

, which 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.

Key Concepts to Understand

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

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If a Poisson distribution is such that P(X=2)=P(X=3)\displaystyle P(X=2) = P(X=3)P(X=2)=P(X=3), then the standard deviation of the distribution is: