Q10

1 markMCQSection A

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

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

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.

Probability Distributions

Poisson Distribution

Options

(A)

m^2

(B)

\sqrt{m}

(C)

m

(D)

\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.

Poisson distributionmean equals varianceparameter mvariancestandard deviation root mdiscrete distributionrare eventsprobability

Marking Scheme

11 mark: correct option (C) m.
2No marks for $m^2$ , $\sqrt{m}$ or $\frac{m}{2}$ .

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.

Analysis & 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, $P(r) = \frac{e^{-m} m^r}{r!}$ .
Direct computation gives $E(X) = m$ and $\text{Var}(X) = m$ , 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) $m^2$ is the square of the mean, not the variance.
(B) $\sqrt{m}$ is the standard deviation of the Poisson distribution (the square root of the variance), not the variance itself.
(D) $\frac{m}{2}$ has no basis in the Poisson formula.

Common Mistakes

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

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.

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Frequently Asked Questions

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

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

Why are mean and variance equal for a Poisson distribution?

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