Skip to main content

Exponential distribution


Exponential Distribution

Probability Density FunctionThe general formula for the probability density function of the exponential distribution isf(x) = (1/beta)*EXP(-(x - mu)/beta) for x >= mu; beta > 0
where mu is the location parameter and beta is the scale parameter(the scale parameter is often referred to as lambda which equals 1/beta). The case where mu = 0 and beta = 1 is called the standard exponential distribution. The equation for the standard exponential distribution is
f(x) = EXP(-x) for x >= 0
The general form of probability functions can be expressed in terms of the standard distribution. Subsequent formulas in this section are given for the 1-parameter (i.e., with scale parameter) form of the function.
The following is the plot of the exponential probability density function.
plot of the exponential probability density function
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the exponential distribution isF(x) = 1 - EXP(-x/beta) for x >= 0; beta > 0
The following is the plot of the exponential cumulative distribution function.
plot of the exponential cumulative distribution
Percent Point FunctionThe formula for the percent point function of the exponential distribution isG(p) = -beta*ln(1-p)
The following is the plot of the exponential percent point function.
plot of the exponential percent point function
Hazard FunctionThe formula for the hazard function of the exponential distribution ish(x) = lambda for x >= 0
The following is the plot of the exponential hazard function.
plot of the exponential hazard function
Cumulative Hazard FunctionThe formula for the cumulative hazard function of the exponential distribution isH(x) = x/beta for x >= 0; beta > 0
The following is the plot of the exponential cumulative hazard function.
plot of the exponential cumulative hazard function
Survival FunctionThe formula for the survival function of the exponential distribution isS(x) = EXP(-x/beta) for x>= 0; beta > 0
The following is the plot of the exponential survival function.
plot of the exponential survival function
Inverse Survival FunctionThe formula for the inverse survival function of the exponential distribution isG(p) = -beta*ln(p) 0 <= p < 1; beta > 0
The following is the plot of the exponential inverse survival function.
plot of the exponential inverse survival function
Common Statistics
Meanbeta
Medianbeta*ln(2)
ModeZero
RangeZero to plus infinity
Standard Deviationbeta
Coefficient of Variation1
Skewness2
Kurtosis9
Parameter EstimationFor the full sample case, the maximum likelihood estimator of the scale parameter is the sample mean. Maximum likelihood estimation for the exponential distribution is discussed in the chapter on reliability (Chapter 8). It is also discussed in chapter 19 of Johnson, Kotz, and Balakrishnan.
CommentsThe exponential distribution is primarily used in reliabilityapplications. The exponential distribution is used to model data with a constant failure rate (indicated by the hazard plot which is simply equal to a constant).
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the exponential distribution

Comments

Post a Comment

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 ...
Post a Comment
Read more

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 commo...
Post a Comment
Read more

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 : Statistica...
Post a Comment
Read more