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Describe the procedure to generate samples from: 1) Erlang distribution. 2) Exponential distribution.

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• The inverse transform technique can be used to sample from the exponential, the uniform, the Weibull, the triangular distributions and form empirical distributions.

• Additionally, it is the underlying principle for sampling from wide variety of discrete distributions.

Erlang distribution

• The pdf given by Equation (5.34) is often referred to as the Erlang distribution of order (or number of phases) k when β = k, an integer.

• Consider a series of k stations that must be passed through in order to complete the servicing of a customer. An. Additional customer cannot enter the first station until the customer in process has negotiated all the stations.

• Each station has an exponential distribution of service time with parameter k θ.

• Equations (5.35) and (5.36), which state the mean and variance of a gamma distribution, arevalid regardless of the value of β.

The mean variance of the gamma distribution are given by

E(X)=$\frac{1}{θ}$

V(X)=$\frac{1}{βθ^2}$

The expected value of the sum of random variables is the sum of the expected value of each random variable.

E(X)=E($X_1$)+E($X_2$)+E($X_3$)+------+E($X_k$)

Thus, When β = k, a positive integer, the cdf given by , giving which is the sum of Poisson terms with mean α= kθx.