What is Gamma Distribution: Formula, Examples & Questions

The gamma distribution is a type of statistical distribution that is related to the beta distribution. It 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. The 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. The gamma distribution formula is:

$f\left(x\right|k,\theta )=\frac{x^{k-1}{e}^{-\frac{x}{\theta }}}{{\theta }^k\Gamma \left(k\right)}$

We will be learning about the gamma distribution formula in a later section. 

The gamma distribution formula is:

$f\left(x\right|k,\theta )=\frac{x^{k-1}{e}^{-\frac{x}{\theta }}}{{\theta }^k\Gamma \left(k\right)}$

Here:

• x>0
• k is shape parameter
• θ is scale parameter
• Γ(k) is the gamma function, which is $\Gamma \left(k\right)={\int }_0^{\infty }{t}^{k-1}{e}^{-t}dt$

In case of an integer value of k, gamma function will be Γ(k)=(k−1)!.

IIT JAM and JEE Main often ask questions related to the Gamma distribution function, which has been discussed below.

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.

The following are the properties of the gamma distribution:

• The gamma distribution is defined by two parameters: k which is the shape parameter (denoted as α) and θ which is the scale parameter. Sometimes, rate parameter β=1/θ is used in place of the scale parameter.
• 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.
• The PDF of gamma distribution for a random variable X is given by: 

$f\left(x\right|k,\theta )=\frac{x^{k-1}{e}^{-\frac{x}{\theta }}}{{\theta }^k\Gamma \left(k\right)}$ where x > 0, and Gamma(k) is the gamma function.

• When shape parameter k is 1, gamma distribution will reduce to exponential distribution with rate parameter 1/θ.
• In the case, when θ=2 and k=n/2 where n is an integer, gamma distribution will be equal to the chi-squared distribution with n degrees of freedom.
• The shape parameter k determines shape of distribution. For k<1, distribution is highly skewed with long tail to right. As k increases, gamma distribution becomes more symmetric and bell-shaped.
• The gamma function Γ(k) is defined as: $\Gamma \left(k\right)={\int }_0^{\infty }{t}^{k-1}{e}^{-t}dt$

Knowing these properties is especially useful while solving NEET and CUET exam questions. No direct questions are asked. Instead, application-based questions are asked in these exams.

If you have to solve some difficult mathematical problems, and for each problem, you are taking about ½ an hour, then it will take you somewhere around 2 to 4 hours to solve four such problems. Especially, IISER and GATE exams ask questions related to the Gamma distribution.

Because you are solving one problem in ½ an hour, thus, θ = 1 / 0.5 = 2, the theorem for each hour on 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. 

Let us take a look at some illustrative examples related to gamma distribution:

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, ∞).