**Properties of poisson distribution :**

Students who would like to learn poisson distribution must be aware of the properties of poisson distribution.

Because, without knowing the properties, always it is difficult to solve probability problems using poisson distribution.

So, let us come to know the 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).

**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

σ = √2.25

σ = 1.5

After having gone through the stuff given above, we hope that the students would have understood "Poisson distribution properties".

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