Poisson Distribution

 How to understand Poisson distribution?

Poisson distribution was developed by a French Mathematician Dr. Simon Denis Poisson in 1837. It is a discrete probability distribution that is widely used in statistical work, It is used in those situations where the probability of happening an event is very small. So the event rarely occurs. For example, for knowing the probability of road accidents, Fire manufacturing defects of occurring earthquakes in a year. This distribution can be obtained by knowing the mean of occurrence of an event. The mean may either be based on the size of the sample or on the basis of past experience. The mean of the Poisson distribution is represented by m(m=mean X=np). The Poisson distribution is a discrete probability distribution used to model the number of events occurring in a fixed interval of time or space when the events are rare and random, but the average rate of occurrence is known. It is named after the French mathematician Siméon Denis Poisson, who introduced it in the 19th century. The Poisson distribution is particularly useful for situations where events occur independently and with a constant average rate. The Poisson distribution is a probability distribution that models the number of events occurring within a fixed interval of time or space, given a known average rate of occurrence. It is named after the French mathematician Siméon Denis Poisson, who introduced the distribution in the early 19th century. The Poisson distribution is particularly useful in situations where events happen randomly and independently with a constant average rate, and it can be used to answer questions like "How many events are likely to occur in a given time period?"

Key characteristics of the Poisson distribution:

Probability Mass Function (PMF): The probability mass function of the Poisson distribution describes the probability of observing a specific number of events (k) in a fixed interval, given the average rate of occurrence (λ). The PMF is defined as follows:

P(X = k) = (e^(-λ) * λ^k) / k!

Where:

P(X = k) is the probability of observing k events.
e is the base of the natural logarithm (approximately 2.71828).
λ (lambda) is the average rate of occurrence within the fixed interval.
k is the number of events you want to find the probability for.
k! represents the factorial of k.
Mean and Variance: The mean (expected value) of a Poisson distribution is equal to λ, and the variance is also equal to λ. This means that both the average number of events and the spread or variability of the distribution are determined by the same parameter, λ.
Independence: The Poisson distribution assumes that events occur independently of one another within the fixed interval. This means that the occurrence of one event does not affect the probability of the occurrence of another event.
Rare Events: The Poisson distribution is particularly applicable when the average rate of occurrence, λ, is relatively small, and the events themselves are rare. In such cases, it approximates the binomial distribution when the number of trials is large, and the probability of success is small.
Common applications of the Poisson distribution include modeling:

The number of phone calls at a call center in a given hour.
The number of accidents at a particular intersection in a day.
The number of emails received in an hour.
The number of customers arriving at a store during a specified time period.
Overall, the Poisson distribution is a valuable tool in probability and statistics for analyzing random events that occur at a consistent, average rate within a fixed interval.
 

Key characteristics and properties of the Poisson distribution include:

Discreteness: The Poisson distribution is a discrete distribution, meaning it describes the probability of observing whole numbers (0, 1, 2, 3, etc.) of events in a given interval.

Rate Parameter (λ): The Poisson distribution is characterized by a single parameter, often denoted as λ (lambda), which represents the average rate of events occurring in the specified interval. λ is a positive real number.

Probability Mass Function (PMF): The probability mass function of the Poisson distribution is given by the formula:

P(X = k) = (e^(-λ) * λ^k) / k!

Where:

P(X = k) is the probability of observing k events.

λ is the average rate of events.

e is the base of the natural logarithm, approximately 2.71828.

k is the number of events (a non-negative integer).

k! represents the factorial of k.

Mean and Variance: The mean (μ) and variance (σ²) of the Poisson distribution are both equal to λ. That is, μ = σ² = λ. This means that the distribution's shape and spread are determined by the average rate parameter λ.

Memorylessness: The Poisson distribution is memoryless, meaning that the probability of an event occurring in a future interval does not depend on the history of events before that interval. This property makes it suitable for modeling rare events that are unrelated to previous occurrences.

Applications: The Poisson distribution is commonly used in various fields, including:

Modeling the number of customer arrivals at a service center in a given time period.

Analyzing the number of accidents at a specific intersection over a year.

Estimating the number of phone calls received at a call center in an hour.

Assessing the number of goals scored in a soccer match.

Analyzing the distribution of rare mutations in DNA sequences.

Relationship to Binomial Distribution: In cases where the number of trials (n) in a binomial distribution is large, and the probability of success (p) is small, the binomial distribution closely approximates the Poisson distribution with λ = np.

Limitation: The Poisson distribution assumes that events are rare and random. It may not be appropriate for situations where events are not independent, the rate of occurrence varies over time, or there is a clustering of events.

In summary, the Poisson distribution is a valuable tool for modeling the random occurrence of rare events with a known average rate. Its simplicity and applicability make it a useful choice in various fields for analyzing count data and making predictions about the number of events in a given interval.

Properties of Poisson distribution

The Poisson distribution has some important  properties:

a) It is a discrete probability distribution in which the number of successes is given in the whole numbers such as 0,1,2,3,..... etc. So the number of successes cannot be in a fraction.

b) It has only one parameter m. It is the meaning of Poisson distribution(m=meanx=np). The entire distribution can be known from this parameter.

c) The shape of the Poisson distribution curve is always positively skewed and leptokurtic(Beta1 is +ve and B2>3). But as its mean value(m) increases, its skewness decreases. If m is very large then the distribution tends to be symmetrical.

d)This distribution is used in those circumstances where the probability is happening an event is very small(p tends to 0) and the number of trials is infinitely large.

e) Poisson distribution is a limiting form of binomial distribution if some conditions are satisfied.

Values Of Poisson Distribution

Mean=m=mean x=np

Variance=m  (mean=variance)

Standard Deviation =square root m

Moment about Mean:

       mu1 = 0

      mu2 = m

     mu 3 = m

     mu4 = m+3m^2

Moment Coefficient of skewness

         Beta1 =( mu3)^2/(mu2)^2 =m^2/m^3=1/m 

[Beta1 is always greater than zero and it is always positively skewed.]

Moment Coefficient of kurtosis:

Beta2=(mu4)/(mu2)^2=3+(1/m)  [Beta2>3 and it is leptokurtic in shape]

Importance of Poisson Distribution

Poisson distribution has great importance in statistical work.

a) It is used in statistical quality control to find out the number of defects in an item.

b) In insurance problems count the number of casualties.

c) In biology count the number of bacteria.

d)To count the number of typing errors per page in a typed material or a page of the book.

e)To count the number of committed by lovers in a year.

f) To count the number of incoming telephone calls in a town.

Poisson Distribution Fitting

p(r)=[(e)^-m]*[(m)^r]/r!

    r= number of success

    m= mean of poission distribution

   e= 2.7183

More

Normal Distribution

ANOVA

Option Greeks

Binomial Distribution