Theoretical DistributionsMCQMTP March 22/RTP Sep 24Question 3519 of 230
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For a Poisson variate X, P(X=2)=3P(X=4)\displaystyle P(X=2) = 3 P(X=4). Then the standard deviation of X is

Options

A2\displaystyle 2
B4\displaystyle 4
C2\displaystyle \sqrt{2}
D3\displaystyle 3
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Correct Answer

✅ Option c — 2\displaystyle \sqrt{2}

All Options:

  • A2\displaystyle 2
  • B4\displaystyle 4
  • C2\displaystyle \sqrt{2}
  • D3\displaystyle 3

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Detailed Solution & Explanation

Let X\displaystyle X follow a Poisson distribution with parameter m\displaystyle m. The probability mass function is given by: P(X=k)=e−mmkk!P(X = k) = \frac{e^{-m} m^k}{k!} We are given the relation: P(X=2)=3P(X=4)P(X = 2) = 3 P(X = 4) Substituting the PMF values into this relation: e−mm22!=3×e−mm44!\frac{e^{-m} m^2}{2!} = 3 \times \frac{e^{-m} m^4}{4!} e−mm22=3×e−mm424\frac{e^{-m} m^2}{2} = 3 \times \frac{e^{-m} m^4}{24} e−mm22=e−mm48\frac{e^{-m} m^2}{2} = \frac{e^{-m} m^4}{8} Since e−m≠0\displaystyle e^{-m} \neq 0 and m>0\displaystyle m > 0, we can divide both sides by e−mm22\displaystyle \frac{e^{-m} m^2}{2}: 1=m24  ⟹  m2=4  ⟹  m=21 = \frac{m^2}{4} \implies m^2 = 4 \implies m = 2 Thus, the mean parameter is m=2\displaystyle m = 2. For a Poisson distribution, the standard deviation is defined as m\displaystyle \sqrt{m}: SD=m=2\text{SD} = \sqrt{m} = \sqrt{2} Hence, **Option C** 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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