Chapter 6
Uniform Distribution
Probability density is given by:
1
𝑓𝑜𝑟 𝛼 < 𝑥 < 𝛽
𝑢(𝑥; 𝛼, 𝛽) = {𝛽 − 𝛼
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝛼, 𝛽 are real constants with 𝛼 < 𝛽.
The mean and the variance:
𝛼+𝛽 1
𝜇= 𝑎𝑛𝑑 𝜎2 = (𝛽 − 𝛼)2
2 12
Gamma Distribution
Probability density is given by:
1 𝑥
𝛼−1 𝑒 −𝛽
𝑥 𝑓𝑜𝑟 𝑥 > 0
𝑔(𝑥; 𝛼, 𝛽) = { 𝛽 𝛼 𝛤(𝛼)
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝛼, 𝛽 > 0.
The mean and the variance:
𝜇 = 𝛼𝛽 𝑎𝑛𝑑 𝜎 2 = 𝛼𝛽2
Some notes:
1. Gamma function:
∞
𝛤(𝛼) = ∫0 𝑦 𝛼−1 𝑒 −𝑦 𝑑𝑦 for 𝛼 > 0
2. 𝛤(𝛼) = (𝛼 − 1) ∙ 𝛤(𝛼 − 1) for 𝛼 > 1
Prove: using integration by parts:
∞ ∞
∫0 𝑢𝑣 ′ 𝑑𝑥 = 𝑢𝑣 − ∫0 𝑢′𝑣 𝑑𝑥 (where u is the simplest function)
3. 𝛤(1) = 1
4. 𝛤(𝛼) = ( 𝛼 − 1)! When 𝛼 is a positive integer
1
5. 𝛤 (2) = √𝜋
When 𝛼 is not a positive integer, the value of 𝛤(𝛼) will have to be looked up in a special table.
Exponential Distribution
Probability density is given by:
1 −𝑥
𝑔(𝑥; 𝜃) = { 𝜃 𝑒 𝑓𝑜𝑟 𝑥 > 0
𝜃
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝜃 > 0.
This is the special case of the Gamma Distribution with 𝛼 = 1 and 𝛽 = 𝜃.
1
Using 𝜃 = we can obtain the waiting time between successes.
𝜆
The mean and the variance:
𝜇= 𝜃 𝑎𝑛𝑑 𝜎2 = 𝜃2
The exponential distribution applies to:
- The occurrence of the first success in a Poisson process
- The waiting time between two successes
Uniform Distribution
Probability density is given by:
1
𝑓𝑜𝑟 𝛼 < 𝑥 < 𝛽
𝑢(𝑥; 𝛼, 𝛽) = {𝛽 − 𝛼
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝛼, 𝛽 are real constants with 𝛼 < 𝛽.
The mean and the variance:
𝛼+𝛽 1
𝜇= 𝑎𝑛𝑑 𝜎2 = (𝛽 − 𝛼)2
2 12
Gamma Distribution
Probability density is given by:
1 𝑥
𝛼−1 𝑒 −𝛽
𝑥 𝑓𝑜𝑟 𝑥 > 0
𝑔(𝑥; 𝛼, 𝛽) = { 𝛽 𝛼 𝛤(𝛼)
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝛼, 𝛽 > 0.
The mean and the variance:
𝜇 = 𝛼𝛽 𝑎𝑛𝑑 𝜎 2 = 𝛼𝛽2
Some notes:
1. Gamma function:
∞
𝛤(𝛼) = ∫0 𝑦 𝛼−1 𝑒 −𝑦 𝑑𝑦 for 𝛼 > 0
2. 𝛤(𝛼) = (𝛼 − 1) ∙ 𝛤(𝛼 − 1) for 𝛼 > 1
Prove: using integration by parts:
∞ ∞
∫0 𝑢𝑣 ′ 𝑑𝑥 = 𝑢𝑣 − ∫0 𝑢′𝑣 𝑑𝑥 (where u is the simplest function)
3. 𝛤(1) = 1
4. 𝛤(𝛼) = ( 𝛼 − 1)! When 𝛼 is a positive integer
1
5. 𝛤 (2) = √𝜋
When 𝛼 is not a positive integer, the value of 𝛤(𝛼) will have to be looked up in a special table.
Exponential Distribution
Probability density is given by:
1 −𝑥
𝑔(𝑥; 𝜃) = { 𝜃 𝑒 𝑓𝑜𝑟 𝑥 > 0
𝜃
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝜃 > 0.
This is the special case of the Gamma Distribution with 𝛼 = 1 and 𝛽 = 𝜃.
1
Using 𝜃 = we can obtain the waiting time between successes.
𝜆
The mean and the variance:
𝜇= 𝜃 𝑎𝑛𝑑 𝜎2 = 𝜃2
The exponential distribution applies to:
- The occurrence of the first success in a Poisson process
- The waiting time between two successes
