PROBABILITY DISTRIBUTIONS ON RANDOM AND DISCRETE VARIABLES
AITZAZ IMTIAZ
Abstract. The following study is based on studying different distributions of probability and how they
will be used to model data.
We begin our formal introduction by variables which can be classified into the following two categories:
Continuous Variable
Can take any value in an interval
Discrete (Random) Variable
Takes specific separate values
As an example we might take a coin toss 3 times. As such we form the three combinations as:
HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T
h (Number of Heads)
P(H = h)
0
1
8
1
3
8
2
3
8
3
1
8
Theorem 0.1. It is generally true in linear cases that P (H = h) = 1 because sum of probabilities equal 1.
Proof. It is evident that:
1 3 3 3
+ + + =1
8 8 8 8
□
For elaboration we can write that P (H > 1) = 83 + 18 .
1. Probability Distribution
1.1. Preliminaries. Recall the following formulae:
P
P
xi fi
∨ P
n
f
P 2
P 2
xi
x
f
i
V ar(x) =
− (x̄)2 ≡ P i − (x̄)
n
f
p
S.D = V ar(x)
x̄ =
Date: February 21, 2025.
1
x