ErlangDistribution

Generate a set of pseudorandom numbers that are Erlang 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 depends only on the parameter k:

As k grows, the distribution becomes symmetric:

Kurtosis depends only on the parameter k:

As k grows, kurtosis nears the kurtosis of NormalDistribution:

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:

Quantile function:

Assume that the delay caused by a traffic signal is exponentially distributed with an average delay of 0.5 minutes. A driver has to drive a route that passes through seven unsynchronized traffic signals. Find the distribution for the delay passing all signals:

Hence the distribution for the sum of 7 independent exponential variables:

Find the probability that traffic signals cause a delay greater than 5 minutes:

Assume that the duration of telephone calls is exponentially distributed. The average length of a telephone call is 3.7 minutes. Find the probability that nine consecutive phone calls will be longer than 25 minutes:

Summing 9 independent phone call durations:

The probability that they last longer than 25 minutes:

Assume that the time delay in a logic element is exponentially distributed and that the average delay is seconds. The longest sequence of logic elements in a combinational logic network is six. Find the probability that delay through all six elements is longer than seconds:

Summing 6 independent delay distributions:

The probability that the delay is greater than :

A device has 3 lifetime phases: A, B, and C. The time spent in each phase follows exponential distribution with a mean time of 10 hours; after phase C, failure occurs. Find the distribution of the time to failure of this device:

Find the mean time to failure:

Find the probability that such a device would be operational for at least 40 hours:

Simulate time to failure for 30 independent devices:

A system starts with 10 devices; one is active and the remaining nine are on standby. The lifetime of each device has ExponentialDistribution with parameter . When a device fails, it is immediately replaced with another device if there is one still available. The lifetime of the system then follows the distribution:

Find the reliability of the system:

Find the average lifetime of this system:

Find the probability that the system will be operational for at least 5000 hours:

Simulate lifetimes of 30 independent runs of such a system:

Parameter influence on the CDF for each :

Erlang distribution is closed under scaling by a positive factor:

ErlangDistribution converges to a normal distribution as k->∞:

Sum of Erlang-distributed variables follows Erlang distribution:

For identically distributed variables:

Relationships to other distributions:

Sum of k variables with ExponentialDistribution is Erlang distributed:

Compare explicit case:

Erlang distribution is a special case of type 3 PearsonDistribution:

Erlang distribution is a special case of GammaDistribution:

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