
In statistics, Poisson Distribution is a discrete probability distribution that considers the number of successes per unit of continuous variable such as time/distance etc. over the course of many units.

Intervals that can be used are time, distance, area, volume etc

For a series of the discrete event, the average time between events is known, but the exact timing of the event is to occur is not know or is random.

The arrival of an event is independent of the previous event.

For example, the average number of visitors coming on a website in a day are 1000 but the number of visitors between 11 am, and 2 pm not known.
Probability mass function or of x events occurring in an interval –
P(x) = (λx eλ )/x!
e – Euler’s number – 2.718
E(X) = µ – Mean expected value
µ = λ = No. of time event occurs / Interval
Example – Number of visitors coming on the website in 1 hour or a number of Ford SUVs between New Jersey and New York.
Probability of occurrence is –
P(x) = (λx eλ )/x!
e – Euler’s number – 2.718
λ – Average Value
A cumulative mass function is the sum of all discrete probabilities –
The probability of viewing fewer than 10 events in Poisson Distribution is :
P(X, x <10) = ∑9i=0 (λx eλ )/x!
(λ0 eλ )/0! + (λ1 eλ )/1!+ ….
Then probabilities of seeing at least one is
1 – the probability of occurrences none.
P(X, x >=1) = 1 – P(X:x=0)
1 – (λ0 eλ )/0! = 1 – eλ

In Poisson Distribution, the probability of success during the small time interval is proportional to the entire length of the interval.

If you know the expected value λ over an hour, then the expected value over 1 minute of that hour is – λminute = λ hour/60
A quick exercise in understanding it better
A store sells on average 10 mobile phones in a week.

What is the probability of the store to sell 9 phones in the coming week?

What is the probability of the store to sell less than 8 phones in the coming week?

What is the probability of the store of not selling any phone on coming Tuesday?
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