Skip to main content

Poisson distribution


Poisson Distribution

Probability Mass FunctionThe Poisson distribution is used to model the number of events occurring within a given time interval.The formula for the Poisson probability mass function is
p(x,lambda) = EXP(-lambda)*lambda**(x)/x!   for x = 0, 1, 2, ...
lambda is the shape parameter which indicates the average number of events in the given time interval.
The following is the plot of the Poisson probability density function for four values of lambda.
plot of the Poisson probability density function
Cumulative Distribution FunctionThe formula for the Poisson cumulative probability function isF(x,lambda) = SUM[EXP(-lambda)*lambda**i/i!]
   where the summation is for i = 0 to x
The following is the plot of the Poisson cumulative distribution function with the same values of lambda as the pdf plots above.
plot of the Poisson cumulative distribution function
Percent Point FunctionThe Poisson percent point function does not exist in simple closed form. It is computed numerically. Note that because this is a discrete distribution that is only defined for integer values of x, the percent point function is not smooth in the way the percent point function typically is for a continuous distribution.The following is the plot of the Poisson percent point function with the same values of lambda as the pdf plots above.
plot of the Poisson percent point function
Common Statistics
Meanlambda
ModeFor non-integer lambda, it is the largest integer less than lambda. For integer lambda, x = lambda and x = lambda - 1 are both the mode.
Range0 to positive infinity
Standard DeviationSQRT(lambda)
Coefficient of Variation1/SQRT(lambda)
Skewness1/SQRT(lambda)
Kurtosis
Parameter EstimationThe maximum likelihood estimator of lambda isltilde = XBAR
where XBAR is the sample mean.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the Poisson distribution.

Comments

Popular posts from this blog

Frequency Polygons

Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the

Lognormal distribution

Lognormal Distribution Probability Density Function A variable X is lognormally distributed if Y = LN(X) is normally distributed with "LN" denoting the natural logarithm. The general formula for the  probability density function  of the lognormal distribution is where   is the  shape parameter ,   is the  location parameter  and  m is the  scale parameter . The case where   = 0 and  m  = 1 is called the  standard lognormal distribution . The case where   equals zero is called the 2-parameter lognormal distribution. The equation for the standard lognormal distribution is Since the general form of probability functions can be  expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the lognormal probability density function for four values of  . There are several common parameterizations of the lognormal distribution. The form given here is from  Evans, Ha

Basics of Sampling Techniques

Population                A   population   is a group of individuals(or)aggregate of objects under study.It is also known as universe. The population is divided by (i)finite population  (ii)infinite population, (iii) hypothetical population,  subject to a statistical study . A population includes each element from the set of observations that can be made. (i) Finite population : A population is called finite if it is possible to count its individuals. It may also be called a countable population. The number of vehicles crossing a bridge every day, (ii) Infinite population : Sometimes it is not possible to count the units contained in the population. Such a population is called infinite or uncountable. ex, The number of germs in the body of a patient of malaria is perhaps something which is uncountable   (iii) Hypothetical population : Statistical population which has no real existence but is imagined to be generated by repetitions of events of a certain typ