 # Poisson Distribution Formula

Poisson distribution is actually another probability distribution formula.  As per binomial distribution, we won’t be given the number of trials or the probability of success on a certain trail. The average number of successes will be given in a certain time interval. The average number of successes is called “Lambda” and denoted by the symbol “λ”.

The formula for Poisson Distribution formula is given below:

$\large P\left(X=x\right)=\frac{e^{-\lambda}\:\lambda^{x}}{x!}$

Here,

$$\begin{array}{l}\lambda\end{array}$$
is the average number
x is a Poisson random variable.
e is the base of logarithm and e = 2.71828 (approx).

### Solved Example

Question: As only 3 students came to attend the class today, find the probability for exactly 4 students to attend the classes tomorrow.

Solution:

Given,
Average rate of value(

$$\begin{array}{l}\lambda\end{array}$$
) = 3
Poisson random variable(x) = 4

Poisson distribution = P(X = x) =

$$\begin{array}{l}\frac{e^{-\lambda} \lambda^{x}}{x!}\end{array}$$

$$\begin{array}{l}\begin{array}{c}P(X = 4)=\frac{e^{-3} \cdot 3^{4}}{4 !} \\ \\P(X = 4)=0.16803135574154\end{array}\end{array}$$