If 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 X follow a Poisson distribution with parameter m . The probability mass function is given by: P(X = k) = -m mkk! Given the condition that P(X = 2) = P(X = 3) : -m m22! = -m m33! -m m22 = -m m36 Since e-m ≠ 0 and m > 0 : 1/2 = m/6 m = 3 Since the variance of the Poisson distribution is equal to the parameter m , we have: Variance = m = 3 The standard deviation of the Poisson distribution is defined as sqrt(Variance) : SD = sqrt(m) = sqrt(3) Hence, Option A is the correct answer.

If 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 $X$ follow a Poisson distribution with parameter $m$. The probability mass function is given by: $$P(X = k) = \frac{e^{-m} m^k}{k!}$$ Given the condition that $P(X = 2) = P(X = 3)$: $$\frac{e^{-m} m^2}{2!} = \frac{e^{-m} m^3}{3!}$$ $$\frac{e^{-m} m^2}{2} = \frac{e^{-m} m^3}{6}$$ Since $e^{-m} \neq 0$ and $m > 0$: $$\frac{1}{2} = \frac{m}{6} \implies m = 3$$ Since the variance of the Poisson distribution is equal to the parameter $m$, we have: $$\text{Variance} = m = 3$$ The standard deviation of the Poisson distribution is defined as $\sqrt{\text{Variance}}$: $$\text{SD} = \sqrt{m} = \sqrt{3}$$ Hence, **Option A** is the correct answer.

Theoretical DistributionsMCQMTP June 2023 Series IIQuestion 3526 of 230

All Questions

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

Options

\displaystyle \sqrt{3}

\displaystyle 3

\displaystyle 6

\displaystyle 9

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Correct Answer

✅ Option a — $\displaystyle \sqrt{3}$

All Options:

A $\displaystyle \sqrt{3}$
B $\displaystyle 3$
C $\displaystyle 6$
D $\displaystyle 9$

Detailed Solution & Explanation

Let

\displaystyle X

follow a Poisson distribution with parameter

\displaystyle m

. The probability mass function is given by:

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

Given the condition that

\displaystyle P(X = 2) = P(X = 3)

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

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

Since

\displaystyle e^{-m} \neq 0

and

\displaystyle m > 0

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

Since the variance of the Poisson distribution is equal to the parameter

\displaystyle m

, we have:

\text{Variance} = m = 3

The standard deviation of the Poisson distribution is defined as

\displaystyle \sqrt{\text{Variance}}

\text{SD} = \sqrt{m} = \sqrt{3}

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 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