This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “Poisson Distribution”.

1. In a Poisson Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by?

a) m = np

b) m = (np)^{2}

c) m = np(1-p)

d) m = p

View Answer

Explanation: For a discrete probability function, the mean value or the expected value is given by

Mean(μ)=\(\sum_{x=0}^n xp(x)\)

For Poisson Distribution P(x)=\(\frac{e^{-m}m^x}{x!}\) substitute in above equation and solve to get µ = m = np.

2. If ‘m’ is the mean of a Poisson Distribution, then variance is given by ___________

a) m^{2}

b) m^{1⁄2}

c) m

d) ^{m}⁄_{2}

View Answer

Explanation: For a discrete probability function, the variance is given by

Variance (v) = \(\sum_{x=0}^n x^2p(x)-\mu^2\)

Where µ is the mean, substitute P(x)=\(\frac{e^{-m}m^x}{x!}\), in the above equation and put µ = m to obtain

V = m.

3. The p.d.f of Poisson Distribution is given by ___________

a) \(\frac{e^{-m}m^x}{x!}\)

b) \(\frac{e^{-m}x!}{m^x}\)

c) \(\frac{x!}{m^xe^{-m}}\)

d) \(\frac{e^m m^x}{x!}\)

View Answer

Explanation: This is a standard formula for Poisson Distribution, it needs no explanation.

Even though if you are interested to know the derivation in detail, you can refer to any of the books or source on internet that speaks of this matter.

4. If ‘m’ is the mean of a Poisson Distribution, the standard deviation is given by ___________

a) \(\sqrt{m}\)

b) m^{2}

c) m

d) ^{m}⁄_{2}

View Answer

Explanation: The variance of a Poisson distribution with mean ‘m’ is given by V = m, hence

Standard Deviation = \(\sqrt{variance} = \sqrt{m}\)

5. In a Poisson Distribution, the mean and variance are equal.

a) True

b) False

View Answer

Explanation: Mean = m

Variance = m

∴ Mean = Variance.

6. In a Poisson Distribution, if mean (m) = e, then P(x) is given by ___________

a) \(\frac{e^{(x-m)}}{x!}\)

b) \(\frac{e^{(m-x)}}{x!}\)

c) \(\frac{x!}{e^{(m-x)}}\)

d) \(\frac{x!}{e^{(x-m)}}\)

View Answer

Explanation: P(x)=\(\frac{e^{-m}m^x}{x!}\)

Put m = e, and get correct solution.

7. Poisson distribution is applied for ___________

a) Continuous Random Variable

b) Discrete Random Variable

c) Irregular Random Variable

d) Uncertain Random Variable

View Answer

Explanation: Poisson Distribution along with Binomial Distribution is applied for Discrete Random variable. Speaking more precisely, Poisson Distribution is an extension of Binomial Distribution for larger values ‘n’. Since Binomial Distribution is of discrete nature, so is its extension Poisson Distribution.

8. If ‘m’ is the mean of Poisson Distribution, the P(0) is given by ___________

a) e^{-m}

b) e^{m}

c) e

d) m^{-e}

View Answer

Explanation: P(x)=\(\frac{e^{-m}m^x}{x!}\)

Put x = 0, to obtain e

^{-m}.

9. In a Poisson distribution, the mean and standard deviation are equal.

a) True

b) False

View Answer

Explanation: In a Poisson Distribution,

Mean = m

Standard Deivation = m

^{1⁄2}

∴ Mean and Standard deviation are not equal.

10. For a Poisson Distribution, if mean(m) = 1, then P(1) is?

a) 1/e

b) e

c) e/2

d) Indeterminate

View Answer

Explanation: P(x)=\(\frac{e^{-m}m^x}{x!}\)

Put m = x = 1, (given) to obtain 1/e.

11. The recurrence relation between P(x) and P(x +1) in a Poisson distribution is given by ___________

a) P(x+1) – m P(x) = 0

b) m P(x+1) – P(x) = 0

c) (x+1) P(x+1) – m P(x) = 0

d) (x+1) P(x) – x P(x+1) = 0

View Answer

Explanation: P(x)=\(\frac{e^{-m}m^x}{x!}\)

p(x+1) = e

^{-1}m

^{x+1}/(x + 1)!

Divide P(x+1) by P(x) and rearrange to obtain (x+1) P(x+1) – m P(x) = 0.

**Sanfoundry Global Education & Learning Series – Probability and Statistics.**

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