Gamma Distribution: Overview, Questions, Preparation

Gamma distribution is widely used in fields such as engineering, science, and business. Gamma distribution is a type of statistical distribution that is in relation to the beta distribution.  

Meaning of Gamma Distribution

Gamma distribution is defined as inverse scale parameters and shape parameters that have a continuous probability distribution. It is also related to an exponential distribution, Erlang distribution, chi-squared distribution, and normal distribution. Gamma function is denoted by 'Γ.'

It also has two free parameters, namely alpha (α), which is the shale parameter, and beta (β), which is the rate parameter. 

β or the scale parameter is used to scale the distribution, and it gives the dimensional data.

Gamma Distribution Function

Γ(y) represents the gamma function. It is an extended version of the factorial function. Therefore, when n∈{1,2,3…} in that case Γ(y) = (n-1).

When alpha (α) is a positive real number, we define Γ(α) as:

• Γ(α) = 0∫∞ ( ya-1 e-y dy), a> 0.
• When a = 1, 1, Γ(1) = 0∫∞ (e-y dy) = 1.
• When the variables are changed y = λz, now we shall use this definition: Γ(α) = 0∫∞ ya-1 eλy dy, here a, λ >0.

Properties of Gamma Distribution

Following are the properties of the gamma distribution:

When α is a positive real number then, 

• Γ(α) = 0∫∞ ( ya-1 e-y dy), when α > 0.
• 0∫∞ ya-1 eλy dy = Γ(α)/λa, when λ >0.
• Γ(α +1) = α Γ(α).
• Γ(m) = (m-1)!, for m = 1,2,3 …
• Γ(½) = √π.

Example of Gamma Distribution 

If you have to solve some difficult mathematical problems and for each problem, you are taking about ½ an hour; therefore, it will take you somewhere around 2 to 4 hours to solve four such problems.

Because you are solving one problem in ½ an hour, thus, θ = 1 / 0.5 = 2, the theorem for each hour on an average. Now, if we use θ = 2 and k = 4, we can conclude our calculation as follows:
 P ( 2 ≤ X ≤ 4 ) = ∑ 4x = 2 x 4 − 1e − x/2Γ (4) 24 = 0.12388. 

Students will learn about the gamma distribution in class 12th. And, the chapter Statistics holds the weightage of 10 marks in the exam. 

1. What is the mean and the variance for gamma distribution?

Solution.

Mean and variance for gamma distribution is given as

E(X) = α/λ, Var(X) = 1/λ2

2. Is gamma function defined as Γ(α) = 0∫∞ xα−1 e−x dx?

Solution.

True, the gamma function is defined as Γ(α) = 0∫∞ xα−1 e−x dx. Gamma function can also be defined as Γ(α+1) = αΓ(α).

3.Is gamma distribution a multivariate distribution?

Solution.

No, the gamma distribution is a univariate distribution, which means it is only defined for x ranging from (0, ∞).

Q: How is gamma distribution useful?

Q: Explain the working of the gamma function. 

Q: How is gamma distribution different from exponential distribution?

Q: What is the meaning of gamma?

Q: Can gamma distribution be described as memoryless?