Gamma Distribution

A debitage concept of statistics

Gamma Distribution

A probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. a two Parameter continuous distribution which follows gamma function is called gamma distribution

```
  It’s a continuous distribution 
```
```
  It’s 2 parameter distribution 
```

It is used to predict the wait time until future events occur. As we shall see the parameterization below, Gamma Distribution predicts the wait time until the k-th (Shape parameter) event occurs.
When a certain procedure consists of α independent steps, and each step takes Exponential (λ) amount of time, then the total time has Gamma distribution with parameters α and λ.

Exmaple :
Users visit a certain internet site at the average rate of 12 hits per minute. Every sixth visitor receives some promotion that comes in a form of a flashing banner. Then the time between consecutive promotions has Gamma distribution with parameters α = 6 and λ = 12.

Gamma related to Others

The gamma, exponential, and Poisson distributions all model different characteristics of a Poisson process. All these distributions can use lambda as a parameter, which represents that average rate of occurrence.

The gamma distribution models the time between events. Time is a continuous variable, and the gamma distribution is, likewise, a continuous probability distribution. Conversely, the Poisson distribution models the count of events within a set amount of time. A count is a discrete variable and the Poisson distribution is a discrete probability distribution.

The gamma and exponential distributions are equivalent when the gamma distribution has a shape value of 1. Remember that the shape value equals the number of events and the exponential distribution models times for one event. Therefore, a gamma distribution with a shape = 1 is the same as an exponential distribution.
For example, a gamma distribution with a shape = 1 and scale = 3 is equivalent to an exponential distribution with a scale = 3

Gamma (1, λ) == Exponential (λ)

Real Life Exmaples

Corona Virus Patients
Load on Server Computer
Waiting Time in Reservation
The amount of rainfall accumulated in a reservoir(Dam).
Flow of items through manufacturing and distribution processes.

Dependencies(Gamma Function)

To define the family of gamma distributions, we first need to introduce a function that plays an important role in many branches of mathematics like solving the hard integrals.

For α>0 , the gamma function is defined by

Most important properties of gamma function is following:

 For any positive integer t, Γ(t) = (t-1)!

```
 For any α>0, Γ(t+1) = (t)Γ(t)
```
```
 Γ(1/2) = √π
```

Probablity Desity Function(PDF)

It is a two-parameter continuous probability distribution.The derivation of which from Gamma Function we will see. The commonly used parameterizations are as follows-

Shape parameter = k and Scale parameter = θ.
Shape parameter α = k and an Inverse Scale parameter β = 1/θ called a Rate parameter. In exponential distribution, we call it as λ (lambda, λ = 1/θ) which is known as the Rate of the Events happening that follows the Poisson process. While k is the number of events until which we are waiting for the expected event to occur.
Shape parameter = k and a Mean parameter μ = k*θ = α/β.

Using the parameters as k (k>0) and θ (λ = 1/θ) where λ is the rate of the event, we can write the PDF of the Gamma Distribution as–

Therefore, a random variable X is eventually denoted b

Constants

Mean

Varience

CDF

MGF

Question 1:

Imagine you are solving difficult Maths theorems and you expect to solve one every 1/2 hour. Compute the probability that you will have to wait between 2 to 4 hours before you solve four of them.

One theorem every 1/2 hour means we would suppose to get θ = 1 / 0.5 = 2 theorem every hour on average. Using θ = 2 and k = 4, Now we can calculate it as follows

Question 2:

Compilation of a computer program consists of 3 blocks that are processed sequentially, one after another. Each block takes Exponential time with the mean of 5 minutes, independently of other blocks.

(a) Compute the expectation and variance of the total compilation time.
(b) Compute the probability for the entire program to be compiled in less than 12 minutes.

Solution

Question

Solution: The total time T is a sum of three independent Exponential times, therefore, it has Gamma distribution with α = 3. The frequency parameter λ equals (1/5) min−1 because the Exponential compilation time of each block has expectation 1/λ = 5 min.

(a) For a Gamma random variable T with α = 3 and λ = 1/5

(b) A direct solution involves two rounds of integration by parts,

Question 3:

Let X have gamma distribution with λ =1/2 and w=1/2.Find the probabilty density function(PDF) of Y=√X.

Differentiate with respect to y to get the pdf of Y

This is for y>=0. The rang is (0,∞)

Question 4:

Let Xj be a sequence of independent, identically distributed random variables. Their common density is

(It is zero for x<0.) Let

(a) Find the mean and varience of Xn.
(b) For n = 1000, the probability that X₁₀₀₀ is in [1,1.1] is approximately given by

find a and b

Solution

Question (a) Solution:

The Xj have a gamma distribution with w=2 and λ=2. So E[Xj]=1 and var(Xj)=1/2. So the mean of Xn is 1 and its varience is 1/2n.

(b) Solution:

Question 5:

Calculate the Survival Time X in weeks of a randomly selected male mouse exposed to 240 rads of gamma radiation has a gamma distribution with α=8 and β=15.

also calculate probability that a mouse survives between 60 and 120 weeks is

Solution:

Question 6:

Let λ>0 and let X₁,X₂,...,X_n be a random sample from the distribution with the probability density function

What is probability distribution of

Solution

Question Solution:

Proof 1:

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

THE END

Download Share Improve Code