Question 101 markMCQSection A

If 'm' is the mean of a Poisson distribution, then its variance is given by :

Probability Distributions
Poisson Distribution

Options

(A)m2m^2
(B)m\sqrt{m}
(C)mm
(D)m2\frac{m}{2}

Official answer

The correct option is (C) m.


Reason

  • A defining property of the Poisson distribution is that its mean and variance are equal.
  • Hence if mean = m, then variance = m.
Answer keywords:Poisson distributionmean equals varianceparameter mvariancestandard deviation root mdiscrete distributionrare eventsprobability

Marking scheme

  1. 1.1 mark: correct option (C) m.
  2. 2.No marks for m2m^2, m\sqrt{m} or m2\frac{m}{2}.

Hint

For a Poisson distribution, mean and variance are the same number.

Quick oral answer

In a Poisson distribution the mean and the variance are both equal to the parameter m, so if the mean is m the variance is also m.

Common mistakes

  1. 1.Confusing variance (m) with standard deviation (m\sqrt{m}).
  2. 2.Writing m2m^2, treating the Poisson like a distribution where variance is the square of the mean.
  3. 3.Assuming variance is half the mean or some fraction of it.

Analysis and explanation

The Poisson distribution is unique in that its single parameter m serves as both mean and variance.


Concept

  • For a Poisson distribution with parameter m, P(r)=e−mmrr!P(r) = \frac{e^{-m} m^r}{r!}.
  • Direct computation gives E(X)=mE(X) = m and Var(X)=m\text{Var}(X) = m, so mean = variance = m.

Why the key is right

  • Option (C) states variance = m, matching the mean-equals-variance property.

Why the distractors are wrong

  • (A) m2m^2 is the square of the mean, not the variance.
  • (B) m\sqrt{m} is the standard deviation of the Poisson distribution (the square root of the variance), not the variance itself.
  • (D) m2\frac{m}{2} has no basis in the Poisson formula.

Interesting facts

The equality of mean and variance is a hallmark test for Poisson data; when sample variance greatly exceeds the mean, statisticians call it 'over-dispersion' and switch to other models.

The Poisson distribution was published by Simeon Denis Poisson in 1837 and is widely used to model rare events such as call arrivals, typing errors, and radioactive decay counts.

Question details

Q10 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Probability Distributions
Topic
Poisson Distribution
Bloom's level
remember
Syllabus topic
Unit: Probability Distributions — Poisson Distribution: Mean and Variance
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Is the standard deviation of a Poisson distribution m or m\sqrt{m}?

The variance is m, so the standard deviation, being the square root of the variance, is m\sqrt{m}. The question asks for variance, which is m, not m\sqrt{m}.

Why are mean and variance equal for a Poisson distribution?

It follows directly from the probability formula P(r)=e−mmrr!P(r) = \frac{e^{-m} m^r}{r!}. Evaluating E(X)E(X) and E(X2)−[E(X)]2E(X^2) - [E(X)]^2 both yield m, a distinctive feature of the Poisson model.