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# ErlangDistribution

 ErlangDistribution represents the Erlang distribution with shape parameter k and rate .
• The probability density for value in an Erlang distribution is proportional to for , and is zero for .
• ErlangDistribution allows k to be any positive integer and to be any positive real number.
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (7)
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:
Closed form for symbolic order:
Closed form for symbolic order:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (5)
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:
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