**1 Answer**

written 5.6 years ago by |

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.