One of the continuous random variable and continuous distribution is the Gamma distribution, As we know the continuous random variable deals with the continuous values or intervals so is the Gamma distribution with specific probability density function and probability mass function, in the successive discussion we discuss in detail the concept, properties and results with examples of gamma random variable and gamma distribution.
Topics on Gamma Distribution
Gamma random variable or Gamma distribution | what is gamma distribution | define gamma distribution | gamma distribution density function | gamma distribution probability density function | gamma distribution proof
A continuous random variable with probability density function
is known to be Gamma random variable or Gamma distribution where the α>0, λ>0 and the gamma function
we have the very frequent property of gamma function by integration by parts as
If we continue the process starting from n then
and lastly the value of gamma of one will be
thus the value will be
cdf of gamma distribution | cumulative gamma distribution | integration of gamma distribution
The cumulative distribution function(cdf) of gamma random variable or simply the distribution function of gamma random variable is same as that of continuous random variable provided the probability density function is different i.e
here the probability density function is as defined above for the gamma distribution, the cumulative distribution function we can write also as
in both of the above formats the value of pdf is as follows
where the α >0, λ>0 are real numbers.
Gamma distribution formula | formula for gamma distribution | gamma distribution equation | gamma distribution derivation
To find the probability for the gamma random variable the probability density function we have to use for different given α >0 , λ >0 is as
and using the above pdf the distribution for the gamma random variable we can obtain by
Thus the gamma distribution formula require the pdf value and the limits for the gamma random variable as per the requirement.
Gamma distribution example
show that the total probability for the gamma distribution is one with the given probability density function i.e
for λ >0, α>0.
using the formula for the gamma distribution
since the probability density function for the gamma distribution is
which is zero for all the value less than zero so the probability will be now
using the definition of gamma function
and substitution we get
Gamma distribution mean and variance | expectation and variance of gamma distribution | expected value and variance of gamma distribution | Mean of gamma distribution | expected value of gamma distribution | expectation of gamma distribution
In the following discussion we will find the mean and variance for the gamma distribution with the help of standard definitions of expectation and variance of continuous random variables,
The expected value or mean of the continuous random variable X with probability density function
or Gamma random variable X will be
mean of gamma distribution proof | expected value of gamma distribution proof
To obtain the expected value or mean of gamma distribution we will follow the gamma function definition and property,
first by the definition of expectation of continuous random variable and probability density function of gamma random variable we have
by cancelling the common factor and using the definition of gamma function
now as we have the property of gamma function
the value of expectation will be
thus the mean or expected value of gamma random variable or gamma distribution we get is
variance of gamma distribution | variance of a gamma distribution
The variance for the gamma random variable with the given probability density function
or variance of the gamma distribution will be
variance of gamma distribution proof
As we know that the variance is the difference of the expected values as
for the gamma distribution we already have the value of mean
now first let us calculate the value of E[X2], so by definition of expectation for the continuous random variable we have
since the function f(x) is the probability distribution function of gamma distribution as
so the integral will be from zero to infinity only
so by definition of the gamma function we can write
Thus using the property of the gamma function we got the value as
Now putting the value of these expectation in
thus, the value of variance of gamma distribution or gamma random variable is
Gamma distribution parameters | two parameter gamma distribution | 2 variable gamma distribution
The Gamma distribution with the parameters λ>0, α>0 and the probability density function
has statistical parameters mean and variance as
since λ is positive real number, to simplify and easy handling another way is to set λ=1/β so this gives the probability density function in the form
in brief the distribution function or cumulative distribution function for this density we can express as
this gamma density function gives the mean and variance as
which is obvious by the substitution.
Both the way are commonly used either the gamma distribution with the parameter α and λ denoted by gamma (α, λ) or the gamma distribution with the parameters β and λ denoted by gamma (β, λ) with the respective statistical parameters mean and variance in each of the form.
Both are nothing but the same.
Gamma distribution plot | gamma distribution graph| gamma distribution histogram
The nature of the gamma distribution we can easily visualize with the help of graph for some of specific values of the parameters, here we draw the plots for the probability density function and cumulative density function for some values of parameters
let us take probability density function as
then cumulative distribution function will be
Description: graphs for the probability density function and cumulative distribution function by fixing the value of alpha as 1 and varying the value of beta.
Description: graphs for the probability density function and cumulative distribution function by fixing the value of alpha as 2 and varying the value of beta
Description: graphs for the probability density function and cumulative distribution function by fixing the value of alpha as 3 and varying the value of beta
Description: graphs for the probability density function and cumulative distribution function by fixing the value of beta as 1 and varying the value of alpha
Description: graphs for the probability density function and cumulative distribution function by fixing the value of beta as 2 and varying the value of alpha
Description: graphs for the probability density function and cumulative distribution function by fixing the value of beta as 3 and varying the value of alpha.
In general different curves as for alpha varying is
Gamma distribution table | standard gamma distribution table
The numerical value of gamma function
known as incomplete gamma function numerical values as follows
The gamma distribution numerical value for sketching the plot for the probability density function and cumulative distribution function for some initial values are as follows
finding alpha and beta for gamma distribution | how to calculate alpha and beta for gamma distribution | gamma distribution parameter estimation
For a gamma distribution finding alpha and beta we will take mean and variance of the gamma distribution
now we will get value of beta as
only taking some fractions from the gamma distribution we will get the value of alpha and beta.
gamma distribution problems and solutions | gamma distribution example problems | gamma distribution tutorial | gamma distribution question
1. Consider the time require to resolve the problem for a customer is gamma distributed in hours with the mean 1.5 and variance 0.75 what would be the probability that the problem resolving time exceed 2 hours, if time exceeds 2 hours what would be the probability that the problem will resolved in at least 5 hours.
solution: since the random variable is gamma distributed with mean 1.5 and variance 0.75 so we can find the values of alpha and beta and with the help of these values the probability will be
P(X>5 | X>2)=(61/13)e-6=0.011631
2. If the negative feedback in week from the users is modelled in gamma distribution with parameters alpha 2 and beta as 4 after the 12 week negative feedback came after restructuring the quality, from this information can restructuring improves the performance ?
solution: As this is modelled in gamma distribution with α=2, β=4
we will find the mean and standard deviation as μ =E(x)=α * β=4 * 2=8
since the value X=12 is within the standard deviation from the mean so we can not say this is improvement or not by the restructuring the quality, to prove the improvement caused by the restructuring information given is insufficient.
3. Let X be the gamma distribution with parameters α=1/2, λ=1/2 , find the probability density function for the function Y=Square root of X
Solution: let us calculate the cumulative distribution function for Y as
now differentiating this with respect to y gives the probability density function for Y as
and the range for y will be from 0 to infinity
The concept of gamma distribution in probability and statistic is the one of the important day to day applicable distribution of exponential family, all the basic to higher level concept were discussed so far related to gamma distribution, if you require further reading, please go through mentioned books. You can also visit out mathematics page for more Topic
A first course in probability by Sheldon Ross
Schaum’s Outlines of Probability and Statistics
An introduction to probability and statistics by ROHATGI and SALEH
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