Gamma Distribution in Probability and Statistics

Gamma Function

The gamma distribution is one of the most widely used distribution systems. Its prominent use is mainly due to its contingency to exponential and normal distributions. It is characterized by mean µ=αβ and variance σ2=αβ2

The gamma function, shown by Γ(x)Γ(x), is an extension of the factorial function to real (and complex) numbers. Specifically, if n∈ {1, 2, 3...}, then

Γ(n)=(n−1)

More generally, for any positive real number αα, Γ(α) is defined as

Γ(α)=∫∞0xα−1e−xdx, for α>0.

Probability density function: The time of wait until the occurrence of the hth Poisson event with a rate of change λ is

P(x)= λ(λx)h-1/(h-1)
For X~ Gamma(K,O), in which K=h and 0=1/ λ, the gamma density function is represented by

xk-1e-x/o/ Γ(k)ok

where

e is any natural number. For example, e = 2.71828

k represents the number of times of occurrence of a particular event. 

In a given situation if the value of K is equal to a positive integer then, Γ(k) = (k − 1) is known to be the gamma function

θ = 1 / λ denotes the mean value of the number of events per time unit, where the mean time between events is λ. For instance, if two phone calls are an event and the mean time between these two phone calls is 2 hours, then the gamma distribution would be θ=1/2=0.5. In case of finding a mean number between calls in the time period of 5 hours then it would be 5 x 1/2=2.5.

x is any random variable

Cumulative Density Function: The gamma cumulative distribution function is denoted by y(k,x/o)/ Γ(k)

where

if k is a positive integer, then Γ(k) = (k − 1) is the gamma function 

Moment generating function: The gamma moment-generating function is M(t)= (1-ot)-k

Expectation: The expected value of a gamma-distributed random variable x is E(X) = ko

Variance: The gamma variance is V ar(X)=Ko2

Gamma Distribution Formula

f(x)= { xp-1e-z/ Γp 

p>0,0<x<infinity

where p and x are a continuous random variable

Gamma Distribution Mean and Variance

If the shape parameter is k>0 and the scale is θ>0, one parameterization has density function

p(x)=xk−1e−x/θkΓ(k)

where the argument, xx, is non-negative. A random variable with this density has a mean kθ and variance kθ2.

An alternative parameterization uses ϑ=1/θϑ=1/θ as the rate parameter (inverse scale parameter) and has a density

p(x)=xk−1ϑke−xϑΓ(k)

Under this choice, the mean is k/ϑk/ϑ and the variance is k/ϑ2.

Gamma Distribution Mean

Gamma Distribution Mean can be determined by the use of two ways: 

Directly

By Expanding the moment generating function

It has another name which is known as the Expected value of Gamma Distribution. E(x)= fo∞e-xxp-1/ Γp x Dx

1/ Γpf0infinity e-xxpdx 

=Γp+1/ Γp 

=p/(p-1) 

=p

Gamma Distribution Graph

The parameters of the gamma distribution describe the shape of the graph. Shape parameter α and rate parameter β are both greater in value than the value of 1.

When α = 1, this denotes the exponential distribution

When β = 1 this denotes the standard gamma distribution

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The scale parameter β is appointed only for the purpose of scaling the distribution. This can be understood by remarking that wherever the random variable x happens to appear in the density of probability, then it gets divided by β. Meanwhile, the scale parameter offers the dimensional data, which is of great importance for working with the “standard” gamma distribution, i.e., with β = 1.

Gamma Distribution Applications

The gamma distribution uses a discipline of various ranges including queuing models, climatology, and financial services. Instances of events that may be modeled by gamma distribution include:

The amount of rainfall accumulated in a water reserve.

The size of loan collective insurance claims

The flow of items through industrial and circulation processes

The workload on web servers

The various forms of Tele-exchange

The gamma distribution is also appointed for the purpose of modelling errors in a multi-level Poisson regression model because the combination of a Poisson distribution and a gamma distribution is a negative binomial distribution.

Gamma Distribution Example

Say, for instance, you are fishing and you predict to catch a fish once every 1/2 hour. Calculate the probability of the time that you will have to wait between 2 to 4 hours before you catch 4 fishes.

One fish every 1/2 hour signifies that we can expect to catch θ = 1 / 0.5 = 2 fish every hour on an average. Using θ = 2 and k = 4, we can compute this as follows: P(2<X<4)= E4x=2 x4-1e-x/2/Γ (4) 24

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Note that for α=1, we can represent as:

Γ(1)=∫∞0e−xdx=1.

Using the alteration of variable x=λy, we can represent the following equation that is often beneficial when using the gamma distribution:

Γ(α)=λα∫∞0yα−1e−λydyfor α,

Also, using integration by parts it can be depicted that

Γ(α+1)=αΓ(α),for α>0.

Note that if the value of α=n, where n is any positive integer, the above equation reduces to

n=n⋅(n−1)

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Properties of the Gamma Function

For any positive real number αα:

1. Γ(α)=∫∞0xα−1e−xdx

2. ∫∞0xα−1e−λxdx=Γ(α)λα, for λ>0;

3. Γ(α+1)=αΓ(α);

4. Γ(n)=(n−1), for n=1,2,3,⋯;

Example 

Using the properties of the gamma function, show that the gamma PDF integrates to 1, i.e., show that for α,λ>0α,λ>0, we have

∫∞0λαxα−1e−λxΓ(α)dx=1.∫0∞λαxα−1e−λxΓ(α)dx=1.
Solution

In this section, we compute the mean and variance for the gamma distribution. To be precise, we find out that if X∼Gamma(α,λ) then

EX=αλ,Var(X)=αλ2.