Some other important continuous distributions

The Gamma Distribution

Gamma distribution has several applications in civil engineering. One of the applications is in rainfall modeling. Variants of the Gamma distribution are widely used in the field of traffic engineering. This distribution is related to the Poisson process and the relationship is explained below. Figure 5.15 shows the probability density functions of the Gamma distribution with different shape parameters. Figure 5.16 shows the frequency distribution of the headway data and the corresponding probability density function of the Gamma random variable.

Figure 5.15: PDF of Gamma distribution for various shape parameters

Figure 5.16: Modeling of headway data using Gamma distribution

Relationship between the Gamma and the Exponential Distribution functions

If the random variable X is the sum of r independent, exponentially distributed random variables, each with parameter λ, then X follows Gamma distribution with parameters r and λ.

The resulting variable X has the following probability density function;

Where, k and λ are the parameters of the Gamma distribution which are always greater than zero. λ is the scale parameter and k is the shape parameter.

Gamma function is defined as follows

for k > 0

When k is the positive integer, then

Mean and the Variance of Gamma Distribution

When r is an integer and also when

And