Special Continuous Distributions I – Gamma and Chi-Square
Special Continuous Distributions I – Gamma and
Chi-Square
Objectives
By the end of this lecture, students should be able to:
Define the gamma distribution and identify its parameters.
Derive and use the probability density function (PDF) of a gamma
distribution.
Use the incomplete gamma function and tables.
Define the chi-square distribution as a special case of the gamma distribution.
Compute probabilities involving gamma and chi-square variables.
Sam Maseno University August 19, 2025
, Special Continuous Distributions I – Gamma and Chi-Square
The Gamma Distribution
Definition:
A continuous random variable X follows a gamma distribution with shape
parameter α > 0 and scale parameter θ > 0, written as:
X ∼ Gamma(α, θ)
The probability density function (PDF) is:
1
f (x; α, θ) = xα−1 e−x/θ , x>0
Γ(α)θα
Here, Γ(α) is the gamma function, defined as:
Z ∞
Γ(α) = tα−1 e−t dt
0
Where:
x is a possible value of the random variable X, with x > 0.
α is the shape parameter of the distribution, with α > 0.
θ is the scale parameter of the distribution, with θ > 0.
Γ(α) is the gamma function,
Sam Maseno University August 19, 2025
, Special Continuous Distributions I – Gamma and Chi-Square
Gamma Distribution
Special Properties:
The mean of the Gamma distribution is:
µ = αθ
The variance of the Gamma distribution is:
σ 2 = αθ2
When α = 1, the Gamma distribution becomes the exponential distribution:
Gamma(1, θ) = Exponential(θ)
Sam Maseno University August 19, 2025
, Special Continuous Distributions I – Gamma and Chi-Square
Gamma Function and Incomplete Gamma
Gamma Function and Incomplete Gamma Function
The gamma function generalizes the factorial function:
Γ(n) = (n − 1)! for positive integers n
Incomplete Gamma Function:
Z x
γ(α, x) = tα−1 e−t dt
0
Cumulative Distribution Function (CDF) of the Gamma Distribution:
γ(α, x/θ)
P (X ≤ x) =
Γ(α)
Values of the incomplete gamma function are typically obtained from tables or
computed numerically.
Sam Maseno University August 19, 2025
Special Continuous Distributions I – Gamma and
Chi-Square
Objectives
By the end of this lecture, students should be able to:
Define the gamma distribution and identify its parameters.
Derive and use the probability density function (PDF) of a gamma
distribution.
Use the incomplete gamma function and tables.
Define the chi-square distribution as a special case of the gamma distribution.
Compute probabilities involving gamma and chi-square variables.
Sam Maseno University August 19, 2025
, Special Continuous Distributions I – Gamma and Chi-Square
The Gamma Distribution
Definition:
A continuous random variable X follows a gamma distribution with shape
parameter α > 0 and scale parameter θ > 0, written as:
X ∼ Gamma(α, θ)
The probability density function (PDF) is:
1
f (x; α, θ) = xα−1 e−x/θ , x>0
Γ(α)θα
Here, Γ(α) is the gamma function, defined as:
Z ∞
Γ(α) = tα−1 e−t dt
0
Where:
x is a possible value of the random variable X, with x > 0.
α is the shape parameter of the distribution, with α > 0.
θ is the scale parameter of the distribution, with θ > 0.
Γ(α) is the gamma function,
Sam Maseno University August 19, 2025
, Special Continuous Distributions I – Gamma and Chi-Square
Gamma Distribution
Special Properties:
The mean of the Gamma distribution is:
µ = αθ
The variance of the Gamma distribution is:
σ 2 = αθ2
When α = 1, the Gamma distribution becomes the exponential distribution:
Gamma(1, θ) = Exponential(θ)
Sam Maseno University August 19, 2025
, Special Continuous Distributions I – Gamma and Chi-Square
Gamma Function and Incomplete Gamma
Gamma Function and Incomplete Gamma Function
The gamma function generalizes the factorial function:
Γ(n) = (n − 1)! for positive integers n
Incomplete Gamma Function:
Z x
γ(α, x) = tα−1 e−t dt
0
Cumulative Distribution Function (CDF) of the Gamma Distribution:
γ(α, x/θ)
P (X ≤ x) =
Γ(α)
Values of the incomplete gamma function are typically obtained from tables or
computed numerically.
Sam Maseno University August 19, 2025
