This document provides information about probability distributions and related concepts. It discusses random variables, probability mass functions of discrete random variables, probability density functions of continuous random variables, and the mean and variance of the binomial, exponential, and normal distributions. Examples are provided to illustrate key concepts such as defining a random variable, determining probabilities of outcomes, and calculating expected values.

1. BRAINWARE UNIVERSITY
Name: SAMPAD KAR
Roll No: BWU/BTA/22/225
Section: D
Group: D2
Course Name: Probability & Statistics
Course Code: BSCM202

2. Contents
• Random Variables
• Probability Distribution of Discrete random variables
• Probability Distribution of Continuous random variables
• Mean and Variance of Binomial Distribution
• Mean and Variance of Exponential Distribution
• Mean and Variance of Normal Distribution

3. Random Variables
A random variable is a variable which represents the outcome of a trial, an
experiment or an event. It is a number which is different each time the trial or event
is repeated.
For Example,
A coin is tossed 3 times simultaneously. Find the Probability of getting 3 heads?
S = {(H,H,,H),(H,H,T),(H,T,T),(T,H,T),(H,T,H),(T,H,H),(T,T,H),(T,T,T)}
P(getting at least 3 heads) =
1
8

4. Probability Distribution of Discrete random variables
A discrete variable is a variable that can "only" take-on certain numbers on the number line.
The Probability Mass Function (PMF) is also called a probability function or frequency function which
characterizes the distribution of a discrete random variable. Let X be a discrete random variable of a
function, then the probability mass function of a random variable X is given by
Px (x) = P( X=x ), For all x belongs to the range of X
It is noted that the probability function should fall on the condition :
•Px (x) ≥ 0 and
•∑xϵRange(x) Px (x) = 1
Example
When we roll a single dice, the possible outcomes are:
1, 2, 3, 4, 5, 6
The probability of each of these outcomes is 16.
If we define the discrete variable X as:
X: the number obtained when rolling a dice.
Then this is a discrete random variable since the sum of the probabilities of each of these possible
outcomes is equal to 1, indeed:
1
6
+
1
6
+
1
6
+
1
6
+
1
6
+
1
6
=1

5. A continuous random variable is a random variable that has only continuous values. Continuous values are
uncountable and are related to real numbers.
The probability density function (pdf) is used to describe the probabilities associated with a continuous random
variable.
Probability Distribution of Continuous random variables

6. Mean and Variance of Binomial Distribution
For binomial distribution,
pmf = P(X=x) = 𝑥
𝑛
𝐶𝑃𝑥
(1 − 𝑃)𝑛−𝑥
Mean = E(x) = 𝑥=1
𝑛
𝑥. 𝑥
𝑛
𝐶𝑃𝑥
(1 − 𝑝)𝑛−𝑥
= 𝑥=1
𝑛
𝑥.
𝑛!
𝑛−𝑥 ! 𝑥!
𝑃𝑥
(1 − 𝑃)𝑛−𝑥
= 𝑥=1
𝑛
𝑥.
𝑛!
𝑛−𝑥 ! 𝑥(𝑥−1)!
𝑃𝑥(1 − 𝑃)𝑛−𝑥
= 𝑥=1
𝑛 𝑛.(𝑛−1)!
(𝑛−1)−(𝑥−1) ! (𝑥−1)!
𝑃. 𝑃𝑥−1
(1 − 𝑃)(𝑛−1)−(𝑥−1)
= nP 𝑥=1
𝑛 (𝑛−1)!
(𝑛−1)−(𝑥−1) ! (𝑥−1)!
𝑃𝑥−1
(1 − 𝑃)(𝑛−1)−(𝑥−1)
= nP 𝑥=1
𝑛
𝑥−1
𝑛−1
𝐶 𝑃𝑥−1
(1 − 𝑃)(𝑛−1)−(𝑥−1)
= 𝑛𝑃(P + (1 − 𝑃))(𝑛−1)
= nP

7. E(x(x-1)) = 𝑥=1
𝑛
𝑥(𝑥 − 1). 𝑥
𝑛𝐶𝑃𝑥(1 − 𝑝)𝑛−𝑥
= 𝑥=1
𝑛
𝑥(𝑥 − 1).
𝑛!
𝑛−𝑥 ! 𝑥!
𝑃𝑥(1 − 𝑃)𝑛−𝑥
= 𝑥=1
𝑛
𝑥(𝑥 − 1).
𝑛!
𝑛−𝑥 ! 𝑥(𝑥−1)(𝑥−2)!
𝑃𝑥(1 − 𝑃)𝑛−𝑥
= 𝑥=1
𝑛 𝑛.(𝑛−1)(𝑛−2)!
(𝑛−1)−(𝑥−1) ! (𝑥−2)!
𝑃2
. 𝑃𝑥−2
(1 − 𝑃)(𝑛−2)−(𝑥−2)
= (𝑛2−𝑛)𝑃2
𝑥=1
𝑛 (𝑛−2)!
(𝑛−2)−(𝑥−2) ! (𝑥−2)!
𝑃𝑥−2(1 − 𝑃)(𝑛−2)−(𝑥−2)
= (𝑛2
−𝑛)𝑃2
𝑥=1
𝑛
𝑥−2
𝑛−2
𝐶 𝑃𝑥−2
(1 − 𝑃)(𝑛−2)−(𝑥−2)
= (𝑛2−𝑛)𝑃2 (P + (1 − 𝑃))(𝑛−2)
= (𝑛2−𝑛)𝑃2
E(𝑥2
) = E(x(x-1)) + E(x)
= (𝑛2
−𝑛)𝑃2
+ nP
Variance = E(𝑥2) – {E(x)}2
= (𝑛2
−𝑛)𝑃2
+ nP – 𝑛2
𝑃2
= nP−𝑛𝑃2
= nP(1-P)

8. Mean and Variance of Exponential Distribution
For exponential distribution,
f(x) =
Mean = E(x) = −∞
0
0𝑑𝑥 + 0
∞
𝑥.λ 𝑒−λ𝑥
𝑑𝑥
= 0 + λ 0
∞
𝑥𝑒−λ𝑥𝑑𝑥
= λ.
𝛤(1)
λ
2
=
1
λ
Now,
E(𝑥2
) = λ 0
∞
𝑥2
𝑒−λ𝑥
𝑑𝑥
= λ.
𝛤(3)
λ
3 =
2
λ
2
{λ 𝑒−λ𝑥
, x > 0
0 , elsewhere
Variance = E(𝑥2) – {E(x)}2
=
2
λ
2 -
1
λ
2
=
1
λ
2

9. Mean and Variance of Normal Distribution
For Normal Distribution,
σ = Standard Deviation
µ = Mean
σ2= Variance
Standard Normal Distribution, N(0,1):
σ = 1 , μ = 0
f(x) =
1
2Π
𝑒−
𝑧2
2 z =
𝑥−𝜇
𝜎

10. THANK YOU