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
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It’s a continuous distribution
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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:
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For any positive integer t, Γ(t) = (t-1)!
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For any α>0, Γ(t+1) = (t)Γ(t)
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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: 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
- 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.
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
Question 6:
Let λ>0 and let X1,X2,...,Xn be a random sample from the distribution with the probability density function
What is probability distribution of
Proof 1:
Using the Properties of the gamma function show that gamma
PDF integrates to 1
,i.e.,show that for α , λ > 0.
