ExponentialDistribution

Generate a set of pseudorandom numbers that are exponentially distributed:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness and kurtosis are constant:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Closed form for symbolic order:

Hazard function of exponential distribution is constant and depends on the parameter :

Quantile function:

A battery has a lifespan that is exponentially distributed with parameter . Find the probability that a random battery has a lifespan of less than 2500 hours:

Compute directly using the CDF:

A relay has an exponentially distributed lifetime with a failure rate of failures per year. In order to estimate warranty costs, estimate the number of relays out of 10,000 that will fail in the first six months of use. The failure rate is also known as the hazard rate:

Hence the lifetime distribution (in years) for relays is:

The probability of failure within the first six months:

The expected number of failures within the first six months:

A product has a time to failure that is exponentially distributed with parameter . Find the product's reliability at 1, 2, and 3 years. Reliability is another name for SurvivalFunction:

What is the product's failure rate at 1, 2, and 3 years? Failure rate is also known as hazard rate:

Suppose the lifetime of an appliance has an exponential distribution with average lifetime of 10 years. Find the appliance lifetime distribution:

Find the probability that a used appliance with years of use will not fail in the next 5 years:

Using the memoryless property of ExponentialDistribution:

Assume the waiting time a customer spends in a restaurant is exponentially distributed with an average wait time of 5 minutes. Find the probability that the customer will have to wait more than 10 minutes:

Find the probability that the customer has to wait an additional 10 minutes, given that he or she has already been waiting for at least 10 minutes (the past does not matter):

The data contains waiting times in days between serious (magnitude at least 7.5 or over 1000 fatalities) earthquakes worldwide, recorded from 12/16/1902 to 3/4/1977:

Fit ExponentialDistribution to the data:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the average number of days between major earthquakes:

Find the probability that two serious earthquakes occur within 100 days:

Simulate times between the next 30 serious earthquakes occurring worldwide:

The fit is almost as good as the one with DagumDistribution:

Waiting times at a receiver for signals coming from four independent transmitters are exponentially distributed with parameters , , , and , respectively. Find the probability that the signal from the third transmitter arrives first to the receiver:

Find the distribution of the waiting time for any signal at the receiver:

Find the average waiting time for any signal at the receiver:

Simulate the waiting time between signals arriving at the receiver for , , , and :

A system is composed of 4 independent components, each with lifespan exponentially distributed with parameter . Find the probability that no component fails before 500 hours:

Directly use SurvivalFunction:

Find the probability that exactly one component will fail in the first 1200 hours:

Directly use CDF and SurvivalFunction:

By using BooleanCountingFunction you can also define the logical condition:

In an optical communication system, transmitted light generates current at the receiver. The number of electrons follows the parametric mixture of Poisson distribution and other distributions, depending on the type of light. If the source uses coherent laser light of intensity , then the electron count distribution is Poisson:

Which is PoissonDistribution:

If the source uses thermal illumination, then the Poisson parameter follows ExponentialDistribution with parameter and the electron count distribution is:

These two distributions are distinguishable and allow the type of source to be determined:

Parameter influence on the CDF for each :

Exponential distribution is closed under scaling by a positive factor:

The variance is the square of the mean:

The minimum of exponential distributions is exponentially distributed:

The minimum of identically distributed variables:

The exponential distribution is memoryless (the past does not matter):

Relationships to other distributions:

BenktanderWeibullDistribution reduces to a truncated ExponentialDistribution:

Shifted ExponentialDistribution is a BenktanderWeibullDistribution:

Exponential distribution is a limit of a scaled BetaDistribution:

Sum of exponentially distributed random variables has ErlangDistribution:

Explicitly compute PDFs for three variables:

ExtremeValueDistribution is a transformation of exponential distribution:

ExponentialDistribution is a special case of GammaDistribution:

The difference of two variates from exponential distribution follows LaplaceDistribution:

Exponential distribution is a transformation of LaplaceDistribution:

LogisticDistribution is a transformation from exponential distribution:

LogisticDistribution is a transformation of exponential distribution:

ParetoDistribution is a transformation of exponential distribution:

And to obtain general case:

Transformation of a ParetoDistribution yields an exponential distribution:

Exponential distribution is a special case of type 3 PearsonDistribution:

PowerDistribution is a transformation of exponential distribution:

Exponential distribution can be obtained from RayleighDistribution:

ExponentialDistribution is a special case of WeibullDistribution:

WeibullDistribution is a transformation of exponential distribution:

Exponential distribution is the limiting distribution of the where has UniformDistribution:

The parametric mixture of PoissonDistribution and exponential distribution follows GeometricDistribution:

KDistribution can be obtained from ExponentialDistribution and GammaDistribution:

HoytDistribution can be obtained from ExponentialDistribution and ArcSinDistribution:

ParetoDistribution can be obtained as a quotient of ExponentialDistribution and ErlangDistribution:

ParetoDistribution can be obtained as a quotient of ExponentialDistribution and GammaDistribution:

ExponentialDistribution is not defined when is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: