Unit 5: Theoretical Distributions
Bernoulli Distribution
A random variable ‘x’ assumes values 1 and 0 with respective probabilities p and q = 1-p is called a
Bernoulli variable.
x
p(x)
1
P
0
q
The distribution can also be written as
𝑝(𝑥) = 𝑝 𝑥 𝑞1−𝑥
Where x=0, 1
Here the occurrence of 1 means success, 0 means failure.
The mean and variance of Bernoulli distribution is
𝐸 (𝑥) = 𝑝
𝑉𝑎𝑟 (𝑥) = 𝑝𝑞
𝑆𝐷(𝑥) = √𝑝𝑞
Mean
Variance
Standard Deviation
Binomial Distribution
A binomial distribution has two constants which are the parameters of the distribution. It is given as
𝑝(𝑥) = 𝑛𝑥 𝐶 × 𝑝 𝑥 × 𝑞 𝑛−𝑥
𝐸 (𝑥) = 𝑛𝑝
𝑉𝑎𝑟 (𝑥) = 𝑛𝑝𝑞
𝑆𝐷 (𝑥) = √𝑛𝑝𝑞
Mean
Variance
Standard Deviation
Recurrence Relation
𝑇𝑥 =
(𝑛 + 1 − 𝑥) 𝑝
× × 𝑇𝑥−1
𝑥
𝑥
Normal Distribution
A normal distribution has the following parameters:



The parameters are µ and ς
The normal variant with parameters µ and ς is denoted as N(µ, ς 2)
The mean and variance of Normal distribution is
𝐸 (𝑥) = 𝜇
𝑉𝑎𝑟(𝑥) = 𝜍 2
𝑆𝐷(𝑥) = 𝜍
Mean
Variance
Standard Deviation

Let ‘X’ be the normal variate with mean µ and standard deviation ς. It is given as
𝑍=
𝑋−𝜇
𝜍
Poisson distribution
The distribution may be obtained under the following conditions:
a) The number of trials ‘n’ is indefinitely large, that is n->∞
b) The constant probability of success for each trial ‘p’is indefinitely small. That is p->∞
c) np=λ
Poisson distribution is given as
𝑝 (𝑥) =
𝑒 −λ λ𝑥
𝑥!
Recurrence Relation
The recurrence relation between two successive Poisson probabilities is given as
𝑝 (𝑥) =
𝜆
× 𝑝 (𝑥 − 1)
𝑥
The recurrence relation between two successive Poisson frequencies is given by
𝜆
𝑇𝑥 = × 𝑇𝑥−1
𝑥
Hyper Geometric Distribution
The following conditions are the characteristics of a hyper geometric distribution
a)
b)
c)
d)
The result of each draw can be classified into one of the two categories.
Probability of success changes on each draw.
Successive draws are dependent as they are made without replacement.
The draw is repeated for a fixed number of times from an infinite population.
The probability mass function is given as
𝑝(𝑥) =
𝑏
𝐶𝑥𝑎 × 𝐶𝑛−𝑥
𝐶𝑛𝑎+𝑏
Chi Square Distribution
1. ‘n’ is the parameter of χ2-distribution
2. The range of χ2 -distribution is (0, ∞)
3. For a χ2-distribution
Mean
Variance
Standard Deviation
Mode
𝑛
𝑉𝑎𝑟 (𝑥) = 2𝑛
𝑆𝐷(𝑥) = √2𝑛
𝑛 − 2 for n > 2
= 0 for n < 2
Student’s t-Distribution
It has the probability density function
𝑓(𝑡) = 𝐾 ×
1
𝑛+1
2
𝑡2
(1 + )
𝑛
1. ‘n’ is a parameter of the distribution
2. (-∞ <t < ∞) is the range of t-distribution
Mean, Median, Mode
Variance
Standard Deviation
Mode
𝑛
0
for n > 2
𝑛−2
𝑆𝐷(𝑥) = √2𝑛
𝑛 − 2 for n > 2
= 0 for n < 2