Skip to main content

Double exponential distribution

Double Exponential Distribution

Probability Density FunctionThe general formula for the probability density function of the double exponential distribution isf(x) = EXP(-|(x-mu)/beta|)/(2*beta)
where mu is the location parameter and beta is the scale parameter. The case where mu = 0 and beta = 1 is called the standard double exponential distribution. The equation for the standard double exponential distribution is
f(x) = EXP(-|x|)/2
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 double exponential probability density function.
plot of the double exponential probability density function
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the double exponential distribution isF(x) = EXP(x)/2 for x < 0, F(x) = 1 - EXP(-x)/2 for x >= 0
The following is the plot of the double exponential cumulative distribution function.
plot of the double exponential cumulative distribution function
Percent Point FunctionThe formula for the percent point function of the double exponential distribution isG(p) = LOG(2*p) for p <= 0.5, G(p) = -LOG(2*(1-p)) for p > 0.5
The following is the plot of the double exponential percent point function.
plot of the double exponential percent point function
Hazard FunctionThe formula for the hazard function of the double exponential distribution ish(x) = EXP(x)/(2-EXP(x)) for x < 0, h(x) = 1 for x >= 0
The following is the plot of the double exponential hazard function.
plot of the double exponential hazard function
Cumulative Hazard FunctionThe formula for the cumulative hazard function of the double exponential distribution isH(x) = -LOG(1 - EXP(x)/2) for x < 0, H(x) = x + LOG(2) for x >= 0
The following is the plot of the double exponential cumulative hazard function.
plot of the double exponential cumulative hazard function
Survival FunctionThe double exponential survival function can be computed from the cumulative distribution function of the double exponential distribution.The following is the plot of the double exponential survival function.
plot of the double exponential survival function
Inverse Survival FunctionThe formula for the inverse survival function of the double exponential distribution isZ(p) = LOG(2*(1-p)) for p <= 0.5; Z(p) = -LOG(2*p) for p > 0.5
The following is the plot of the double exponential inverse survival function.
plot of the double exponential inverse survival function
Common Statistics
Meanmu
Medianmu
Modemu
RangeNegative infinity to positive infinity
Standard DeviationSQRT(2)*beta
Skewness0
Kurtosis6
Coefficient of VariationSQRT(2)*(beta/mu)
Parameter EstimationThe maximum likelihood estimators of the location and scale parameters of the double exponential distribution aremuhat = XMEDIAN
betahat = SUM[|X(i) - XMEDIAN|]/N where the summation is from 1 to N
where XMEDIAN is the sample median.
SoftwareSome general purpose statistical software programs support at least some of the probability functions for the double 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