PROPERTIES OF POISSON DISTRIBUTION

1. n the number of trials is indefinitely large. That is,

n → ∞

2. p the constant probability of success in each trial is very small. That is,

p → 0

3. Poisson distribution is known as a uni-parametric distribution as it is characterized by only one parameter m. 

4. The mean of Poisson distribution is given by m. That is, 

μ = m

5.  The variance of the Poisson distribution is given by

σ² = m

6.  Like binomial distribution, Poisson distribution could be also uni-modal or bi-modal depending upon the value of the parameter m.

m is a non integer ----> Uni-modal

Here, the mode  =  the largest integer contained in m

m is a integer ----> Bi-modal 

Here, the mode = m, m - 1

7. Poisson approximation to Binomial distribution :

If n, the number of independent trials of a binomial distribution, tends to infinity and p, the probability of a success, tends to zero, so that m = np remains finite, then a binomial distribution with parameters n and p can be approximated by a Poisson distribution with parameter m (= np).

In other words when n is rather large and p is rather small so that m = np is moderate then

(n, p) ≅ P(m)

8. Additive property of binomial distribution.

Let X and Y be the two independent Poisson variables. 

X is having the parameter m₁

and

Y is having the parameter m₂

Then (X + Y) will also be a Poisson variable with the parameter (m₁ + m₂). 

Practice Problems

Problem 1 :

If the mean of a Poisson distribution is 2.7, find its mode.

Solution : 

Given : Mean = 2.7

That is, m = 2.7 

Since the mean 2.7 is a non integer, the given Poisson distribution is uni-modal. 

Therefore, the mode of the given Poisson distribution is

= Largest integer contained in m

= Largest integer contained in 2.7

= 2

Problem 2 :

If the mean of a Poisson distribution is 2.25, find its standard deviation. 

Solution : 

Given : Mean = 2.25

That is, m = 2.25 

Standard deviation of the Poisson distribution is given by 

σ = √m

= √2.25

= 1.5

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