Probability and Statistics Questions and Answers – Poisson Distribution

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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

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) m2
b) m12
c) m
d) m2

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) $$\frac{e^m m^x}{x!}$$

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.
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4. If ‘m’ is the mean of a Poisson Distribution, the standard deviation is given by ___________
a) $$\sqrt{m}$$
b) m2
c) m
d) m2

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

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

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

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) em
c) e
d) m-e

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

Explanation: In a Poisson Distribution,
Mean = m
Standard Deivation = m12
∴ 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

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

Explanation: P(x)=$$\frac{e^{-m}m^x}{x!}$$
p(x+1) = e-1 mx+1 /(x + 1)!
Divide P(x+1) by P(x) and rearrange to obtain (x+1) P(x+1) – m P(x) = 0.

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