Theoretical DistributionsMCQPYQ June 22Question 3498 of 230
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If Standard Deviation is 1.732 then what is the value of poisson distribution. The P(−2.48<X<3.54)\displaystyle P(-2.48 < X < 3.54) is

Options

A0.73
B0.65
C0.86
D0.81
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Correct Answer

✅ Option b — 0.65

All Options:

  • A0.73
  • B0.65
  • C0.86
  • D0.81

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Detailed Solution & Explanation

For a Poisson distribution, the standard deviation is given by m\displaystyle \sqrt{m}, where m\displaystyle m is the mean parameter. Given that the standard deviation is 1.732\displaystyle 1.732: m=1.732≈3  ⟹  m=3\sqrt{m} = 1.732 \approx \sqrt{3} \implies m = 3 We need to find the probability: P(−2.48<X<3.54)P(-2.48 < X < 3.54) Since the Poisson random variable X\displaystyle X can only take non-negative integer values (0,1,2,3,…\displaystyle 0, 1, 2, 3, \dots), the range −2.48<X<3.54\displaystyle -2.48 < X < 3.54 corresponds to: X∈{0,1,2,3}X \in \{0, 1, 2, 3\} Thus, we need to calculate P(X≤3)\displaystyle P(X \le 3): P(X≤3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)P(X \le 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) P(X=k)=e−33kk!P(X = k) = \frac{e^{-3} 3^k}{k!} Using the value e−3≈0.049787\displaystyle e^{-3} \approx 0.049787: P(X=0)=e−3≈0.049787P(X = 0) = e^{-3} \approx 0.049787 P(X=1)=3e−3≈0.149361P(X = 1) = 3 e^{-3} \approx 0.149361 P(X=2)=92e−3=4.5e−3≈0.224042P(X = 2) = \frac{9}{2} e^{-3} = 4.5 e^{-3} \approx 0.224042 P(X=3)=276e−3=4.5e−3≈0.224042P(X = 3) = \frac{27}{6} e^{-3} = 4.5 e^{-3} \approx 0.224042 Summing these probabilities: P(X≤3)=e−3(1+3+4.5+4.5)=13e−3≈13×0.049787=0.647231≈0.65P(X \le 3) = e^{-3} (1 + 3 + 4.5 + 4.5) = 13 e^{-3} \approx 13 \times 0.049787 = 0.647231 \approx 0.65 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

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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