Guide to Probability & Distributions: From Fundamentals to Advanced Concepts

E-mail id -
paras0101m@gmail.com

"50% Chance This Title
is Accurate: Exploring
Probability!"

What is Probability?
Measure of the
likelihood of
events(chance)
Values between
0 (impossible)
and 1 (certain)
Probability is a numerical measure of the likelihood that a specific event
will occur, defined as the ratio of the number of favorable outcomes to
the total number of possible outcomes.

Examples
 🎯 The chance of hitting a bullseye in
darts depends on your aim — that’s
probability in action!
 🍕 When you grab a random slice of
pizza, the odds of getting the one
with extra cheese is pure luck —
and probability.
 🚦 The probability of hitting a red light
on your way to work shows how life
loves randomness.
 Impossible Event
chances tomorrow I came office in
dinosaur
 Sure Event
the Sun will rise tomorrow in India

IMPORTANT PROBABILITY SIGNS, NAMES &
MEANINGS
Symbol Name Meaning / Usage Example
P(A) Probability of A Chance of event A occurring P(Rain) = 0.7
∪ Union A or B occurs (either or both) A B
∪
∩ Intersection Both A and B occur A ∩ B
A' or Aᶜ Complement of A A does not occur P(A') = 1 - P(A)
⊂ Subset A is a part of B A B
⊂
= Equal Both sides are the same P(A) = 0.4
≠ Not Equal Both sides are not the same P(A) ≠ P(B)
⊆ Subset or Equal A is a subset or equal to B A B
⊆
∅ Null / Empty Set No outcome (impossible event) A ∩ B = if mutually exclusive
∅
S Sample Space Set of all possible outcomes S = {1,2,3,4,5,6} for dice roll
→ Implies If A then B A → B
⇔ If and Only If (iff)
A happens if and only if B
happens
A B
⇔

Sample Space & Events
Sample Space
(Ω): All possible
outcomes
Event (A): A
subset of Ω
Sample Space: The set of all possible outcomes of a random
experiment.
Event: A subset of the sample space representing one or more
outcomes.

TYPES OF SAMPLE SPACES IN
PROBABILITY
Type Description
Example (Better & Fun Real-
Life)
Discrete / Finite
Countable and limited
outcomes
Days you wear clean socks in a
week: {Mon, Wed, Fri}
Discrete / Infinite
Countable but never-ending
values
Number of customers arriving
at a bank in a day: {0, 1, 2, 3,
…}
Continuous(intervals)
Infinite and uncountable
outcomes
Time it takes to finish a cup of
coffee : S = (0, ∞) minutes
☕

CONDITIONS THAT ELEMENTS OF SETS
MUST FOLLOW :
 1. Mutually Exclusive
Events that cannot happen together.
🔹 Example: A traffic light is either Red, Yellow, or Green — never more than one at a
time.
🔹 Mathematically: P(A ∩ B) = 0
 2. Collectively Exhaustive
Together they cover all possibilities — at least one must happen.
🔹 Example: A tossed coin must land as either Heads or Tails.
🔹 Mathematically: A B = Sample Space (S)If E1,E2,...,En are collectively exhaustive
∪
events, then: P(E1 E2 … En)=1
∪ ∪ ∪
 3. Right Granularity(Balanced Details,Depend on situations)
Choose a level of detail that is neither too broad nor too complex.
🔹 Too lack: Weather = Rain or No Rain — lacks depth
🔹 Too Fine: Rain at 3:00 PM, Drizzle at 3:02 PM, etc. — too messy
🔹 Right Granularity: Morning Rain, Afternoon Rain, No Rain — just enough detail for good
decisions

Axioms of
Probability(Rules)
 Non-negativity: P(A) ≥ 0
 Normalization: P(Ω) = 1
 Finite Additivity:
P(A u B) = P(A) + P(B) for disjoint
A, B

Consequence of Axioms
 Basic Consequences of Probability Axioms
1. P(A) ≥ 0 → Probabilities are never negative
2. P(S) = 1 → Total probability of entire sample space is 1
3. P( ) = 0
∅ → Probability of impossible event is 0
4. P(A B) = P(A) + P(B)
∪ if A and B are disjoint
5. P(A) + P(A′) = 1 → An event and its complement add up to 1

Types of Probability
Classical
Probability
Empirical
(Experimental)
Probability
Subjective
Probability
AXIOMATIC PROBABILITY

Classical Probability
Based on symmetry
and equally likely
outcomes
Formula: P(A) =
Number of
favorable outcomes
/ Total outcomes
Probability based on equally likely outcomes in a known sample
space.

💊 Example: Choosing a Pill
A doctor has 4 identical-looking pills in a
box:
 1 is a painkiller (needed),
 3 are vitamins (not harmful).
If the doctor randomly picks one pill without
looking, what's the probability it’s the
painkiller?
Total pills = 4
Favorable (painkiller) = 1

✅ This is classical probability because:
Each pill is equally likely to be chosen.
All outcomes are known and equally likely.

Empirical Probability
Based on
observed
frequency
Formula: P(A) =
Frequency of
A / Total trials
Probability based on past data or observation on anexperiments.

Example
 📊 Empirical Probability (Based on Data/Observation)
 💉 Example: Reaction to a Vaccine
 Out of 1000 patients who received a new vaccine, 50 experienced mild
side effects.
 ✅ This is empirical probability because it's based on observed data, not
assumptions.

Subjective Probability
Based on
personal
judgment or
belief
Useful when
data or
symmetry is
absent
Probability based on personal judgment, belief, or intuition
rather than data

Example(Personal
judgement)
 ☁️Subjective Probability Example – Rain:
 "I think there's a 70% chance it will rain today — because the
sky is cloudy, , and it always rains around this time in July."
 🔍 Why is this subjective probability?
 It’s not based on formal weather data or mathematical
models.
 It’s based on the person’s experience, intuition, and beliefs.
 Different people might assign different probabilities for the
same event.

Axiomatic Probability(proper rules)
defines probability
using a set of logical
rules(AXIOMS)
Ensures all probabilities
are between 0 and 1,
and the total
probability of all
possible outcomes is 1.
Probability defined using a set of formal rules (axioms) by
Kolmogorov.

Example
Let A = “A randomly selected manufactured widget is defective,” with P(A) = 0.1.
Let B = “It is non defective,” with P(B) = 0.9.
‑
 Explanation
 0 ≤ P(A), P(B) ≤ 1 (both 0.1 and 0.9 lie in [0, 1])
 A and B are mutually exclusive (a widget can’t be both
defective and non defective)
‑
 P(A B) = P(A) + P(B) = 0.1 + 0.9 = 1
∪
 These satisfy all three of Kolmogorov’s axioms, illustrating
axiomatic probability in a manufacturing context

Venn Diagrams
Visual
representation
of events
Areas
correspond to
probabilities
Useful for
union,
intersection,
set difference

Q) A class contains 100 students; 70 of them like mathematics, 60
like physics, and 40 like both. If a student is chosen at random,
using a Venn diagram, find the probability that they like
mathematics but not physics.

Guide to Probability & Distributions: From Fundamentals to Advanced Concepts

From Observation to Smart Decisions —
With the Help of Probability!
 Let’s say you want to predict rain at the park:
 🌦 Observe the weather daily
→ You note down if it’s sunny, cloudy, or rainy.
 📊 Analyze with statistics
→ You find patterns, like “rain often follows clouds.”
 🧠 Build a probability model
→ For example, “Cloudy → 70% chance of rain.”
 📅 Make predictions
→ Use the model to guess tomorrow’s weather.
 ✅ Test and improve
→ Check if your guess was right, and adjust.

Addition Rule
Union of events: P(A B) = P(A) +
∪
P(B) - P(A ∩ B)
Special case for disjoint: P(A B) =
∪
P(A) + P(B)
We use addition in probability to find the chance of either of multiple
events happening (union), especially when events can overlap.

Example
 ☂️(Mohali Monsoon )
 A: It rains in Mohali → P(A)=0.5
 B: You forget your umbrella → P(B)=0.3
 Both happen together → P(A∩B)=0.1

Conditional
probability
Conditional probability is the
chance of an event(A)
happening given that another
event(B) has already
happened.

“In a company training program, 50% of employees enroll
in an advanced workshop, and 45% of all employees
successfully complete it. What is the probability that an
employee completes the workshop given they enrolled?”
 Let:
 B = “Employee enrolls in the workshop” → P(B) = 0.50
 A = “Employee completes the workshop” → P(A ∩ B) = 0.45
 Interpretation:
 Given that an employee enrolled in the workshop, there is a
90% chance they will complete it.

Multiplication Rule
(For dependent events):
P(A ∩ B) = P(A)·P(B|A)OR
P(B).P(AIB)
For independent events:
P(A ∩ B) = P(A)·P(B)
We use the multiplication rule when we want to find the probability
that two or more events happen together (i.e., "and" condition).

Q) You are applying for a job at a company. To get selected, you must first clear
a written test and then pass a personal interview. The probability that you clear
the written test is 0.8. If you clear the written test, the probability that you pass
the interview is 0.6.
What is the probability that you will clear both the written test and the
interview?
 Solution (Using Multiplication Rule):
 Let:
 A: Event of clearing the written test → P(A)=0.8
 B: Event of clearing the interview given the test is
cleared → P(B A)=0.6
∣
 P(A∩B)=P(A)×P(B A)=0.8×0.6=0.48
∣
 There is a 48% probability that you will clear both
the written test and the interview.

• EXAMPLE
In a school, the probability that a student revises before an exam is 0.8.
The probability that a student passes the exam given they revised is 0.9.
• What is the probability that a student both revises and passes the exam?
 🎯 Interpretation:
 There is a 72% chance that a student both revises and passes the exam.

Joint Probability
Probability of
simultaneous
occurrence: P(A
∩ B)
Calculated using
multiplication
rule
Joint Probability is the probability of two (or more) events happening at
the same time.

Dependence of
Events
A and B
dependent events
P(A∩B)=P(A)×P(BIA)
or
P(A∩B)=P(B)x(AIB).

Independence
of Events
A and B
independent if P(A
∩ B) = P(A)·P(B)
Then P(B|A) = P(B)

Q) A person is planning their day. The weather forecast says there's a
30% chance of rain. The person also has a scheduled COVID test, and
the chance of testing positive is 5%. Assuming the weather and health
status are independent events, what is the probability that it rains
today and the person tests positive for COVID?
 Let:
 A = It rains today → P(A)=0.3
 B = The person tests positive for COVID → P(B)=0.05
 These events are independent — the weather doesn't affect the test
result.
 Using Multiplication Rule:
 P(A∩B)=P(A)×P(B)=0.3×0.05=0.015
The probability that it rains today and the person tests positive for
COVID is 0.015 or 1.5%.

Law of Total Probability(Divide and
conquer)
If {Ai} partitions Ω:
P(B) =
Σ P(B|Ai)·P(Ai)
The Law of Total
Probability is used to
find overall
probability of an
event.
The Law of Total Probability helps compute the probability of an event
by considering all possible ways (via a partition of the sample space)
that the event can occur.

• Venn Diagram:
A visual representation using
overlapping circles to show all possible
logical relationships between different
sets.
• Tree diagram
A branching diagram that maps out
all possible outcomes of an event in
a structured, hierarchical way.
Formula
where I =1,2,3,….....,n
no of partition

Q)
A hospital uses three different labs (Lab A, Lab B, and Lab C) to process blood samples. 50% of the
samples go to Lab A, 30% to Lab B, and 20% to Lab C. The probability that a sample is
contaminated is:
• 2% if processed by Lab A
• 5% if processed by Lab B
• 1% if processed by Lab C
What is the overall probability that a randomly selected blood sample is contaminated?
Let:
 C: Event that the blood sample is contaminated
 A,B,C: Events that the sample is sent to Lab A, B, or C respectively
USING LAW OF TOTAL PROBABILITY

Bayes' Theorem(reverse
conditional probability)
Bayes' Theorem is a mathematical formula used to update the probability
of a hypothesis (event) based on new evidence or information.
Where:
• P(A) = Prior probability (initial belief about event A)
• P(B A) = Likelihood (how likely B is if A is true)
∣
• P(B) = Total probability of evidence B
P(A/B)=Posterior probability(new probability after watching evidence

In a city, 1% of people are infected with a rare disease. A diagnostic test detects the
disease correctly 99% of the time (i.e., it gives a positive result if a person has the disease),
but also gives a false positive in 2% of healthy individuals. If a person tests positive, what is
the probability that they actually have the disease?
Q)

CONDITIONAL INDEPENDENCE
Two events A and B are said to be conditionally independent given a
third event C if the occurrence of A and B are independent when C is
known to have occurred.
Mathematically:
P(A∩B C)=P(A C) P(B C)
∣ ∣ ⋅ ∣
Knowing event C has occurred, the information about event A does not
affect the probability of B, and vice versa.
P(A∩B C)=P(A C) P(B C)
∣ ∣ ⋅ ∣

Among obese patients, 40% have high blood pressure and 30%
have high cholesterol. If these conditions are conditionally
independent given obesity, what is the probability that an obese
patient has both?
There Is 12% chance that person has both High BP and High Cholesterol
given he is obese
Q)

INDEPENDENCE OF
COLLECTION OF EVENTS
Independence of a collection of events means that every event in the
group is Independent of any combination of the others.
In other words, the occurrence of one or more events does not affect the
probability of any other event in the collection.

A washing machine, refrigerator, and microwave work
independently.
If the probability that each one works is 0.9, what is the
probability that all three work?
Since the events are independent:
P(A∩B∩C)=P(A) P(B) P(C)=0.9×0.9×0.9=0.729
⋅ ⋅
There is a 72.9% chance that all three appliances work.
Event A: The washing machine works
Event B: The refrigerator works
Event C: The microwave works
Q)

A working professional may or may not get a call from their boss
(event A), it may rain in the morning (event B), and they might
skip breakfast (event C).
Assume these events occur due to unrelated causes.
Are the events pairwise independent?
Q)

defination
Counting in probability refers to the method of systematically
determining the total number of possible outcomes in a situation,
which is essential for calculating probabilities.
Real-life Example:
You’re picking an outfit: you have 3 shirts and 2 trousers.
Using counting, the total number of possible outfits is:
3 shirts×2 trousers=6 outfits

Example :
Imagine you’re customizing a coffee order with:
• 3 types of coffee beans (Arabica, Robusta, Blend)
• 4 milk options (Whole, Skim, Soy, Almond)
• 2 sweeteners (Sugar, Honey)
The total number of unique drinks you can create is:
3×4×2 = 24.

Permutation (order matters)
 Definition:
A permutation is an arrangement of items in a specific order.
Order matters in permutation.
 Real-life Example:
You’re assigning gold, silver, and bronze medals to 3 finalists out of 5
runners.
The number of ways to assign the 3 positions is:
 P(5,3)=5!/(5−3)!=60 ways
 Because who gets which medal matters, it’s a permutation.

In a race with 8 runners, how many ways
can you award Gold, Silver, and Bronze
medals?(without replacement)
 Gold–Silver–Bronze is different from Silver–Gold–Bronze, just like ABC ≠
BAC.

Combinations(Only-Choose)
 Definition:
A combination is a selection of items without caring about the
order.
Order does not matter in combination.
 Real-life Example:
You’re forming a 3-person committee from 5 volunteers.
The number of ways to choose 3 people is:
 Because it doesn’t matter who’s picked first, second, or third —
only who is selected — it’s a combination.
NO SYSTEMATIC ARRANGEMENT

A committee of 3 persons is to be constituted from a group
of 2 men and 3 women. In how many ways can this be
done? How many of these committees would consist of 1
man and 2 women?
 Here, order does not matter. Therefore, we need to count
combinations. There will be as many committees as there are
combinations of 5 different persons taken 3 at a time.
 Now, 1 man can be selected from 2 men in 2C1 ways and 2
women can be selected from 3 women in 3C2 ways.
Therefore, the required number of committees
Q)

Binomial Probability
 Definition: Models the number of “successes” in n independent
trials, each with two outcomes (success/failure) and constant
success probabilityp.
 Key Conditions:
 A fixed number of trials, n.
 Each trial has exactly two outcomes (success or failure).
 Trials are independent.
 Probability of success, p, is the same on every trial.

1. Definition
• Partition n distinct objects into r labeled groups of sizes n1,n2,nr (where
∑ni=n).
• Order of groups matters; order within each group does not.

Multinomial probability
 Generalizes the binomial to experiments with m possible
categories per trial, each with a fixed probability, over n
independent trials.
 Key Conditions:
 A fixed number of trials, n.
 Each trial results in exactly one of m categories.
 Trials are independent.
 Probabilities (p1,p2,…,pm) sum to 1 and stay constant.

Random Variables
Discrete vs
Continuous
Mapping
outcomes to
numerical values
A random variable is a numerical function that assigns a real
number to each outcome in a sample space of a random
experiment.

Random Variables(RV)
Random Variable Definition Examples
1. Discrete RV
Takes on a countable
set of values (often
integers).
– Number of children in a
family
– Number of heads in 10
coin tosses
– Number of customers
arriving today
2. Continuous RV
Takes on any value in an
interval (uncountably
many).
– Amount of rain (mm) in
a day
– Height of a randomly
chosen person (cm)
– Time to complete an
exam (minutes)

Probability Distributions
 In probability theory, a distribution (or probability distribution) is a
function or rule that describes how the probabilities are assigned
to different possible outcomes of a random variable.
 Importance of Distribution in Probability:
 Helps calculate probabilities of events.
 Forms the basis for statistical analysis.
 Used in decision-making under uncertainty.
 Essential in modeling real-world situations like weather, traffic,
stock prices, etc.

NOTE
A probability distribution and a frequency distribution are similar because both show how
values are distributed across a variable. A frequency distribution is based on observed
data, while a probability distribution is theoretical. Relative frequencies approximate
probabilities as sample size increases.
The CDF (Cumulative Distribution Function) gives the probability that a variable is less than
or equal to a value: F(x)=P(X≤x). It summarizes the entire distribution.

Distribution Description Real-Life Example
Bernoulli
Only two possible
outcomes: success (1) or
failure (0).
Tossing a coin once
Binomial
Number of successes in
a fixed number of
independent Bernoulli
trials.
Number of heads in
10 coin tosses
Geometric
Number of trials needed
for the first success.
Flipping a coin until
first head
Poisson
Number of events in a
fixed interval of time or
space.
Number of
customer calls per
hour
Discrete Uniform
All outcomes are
equally likely.
Rolling a fair 6-sided
die
Discrete Probability Distributions

Distribution Description Real-Life Example
Continuous Uniform
All values within a given
interval are equally
likely.
Waiting time
between 1 to 5
minutes
Normal (Gaussian)
Bell-shaped curve; most
values cluster around
the mean.
Heights, exam scores
Exponential
Time between events in
a Poisson process.
Time between two
bus arrivals
Gamma
Generalization of
exponential; models
waiting time for multiple
events.
Time until a machine
fails after multiple
uses
Continuous Probability Distributions

What is PMF(PROBABILITY MASS
FUNCTION)
A Probability Mass Function (PMF) assigns, for each possible
outcome of a discrete random process, the chance that exact a
particular outcome occurs.
A probability mass function (PMF) is simply a table or rule that
tells you, for each possible value a discrete random quantity can
take (like rolling a 1, 2, 3, etc.), exactly how likely that value is—
always giving numbers between 0 and 1 that add up to 1.

Probability Mass Function (PMF)
of a Discrete Random Variable

Joint PMF and Multiple
Random variable

Let’s say we randomly pick a student.
Let:
• X: Score in Math (can be 0, 1)
• Y: Score in English (can be 0, 1)
(1 = Pass, 0 = Fail)
QUESTION

Expected Value & Variance
E(X) = Σ
x·P(X=x)
Var(X) =
E(X²) - [E(X)]²

Expectation
 Expectation is the average value we expect from a random process if
we repeat it many times.
 Think of it like the center of gravity of a probability distribution
 .

Elementary Properties of
Expectation

Properties(Linearity of
Expectation)

Total Expectation
Theoram(divide and
conquer)
if you want to find the overall expected value of
something uncertain, you can break it down into parts
based on known conditions, calculate the expected
value for each part, and then take the weighted
average

variance
Variance tells us how much the values spread out from the average.
A small variance means most values are close to the mean,
while a large variance means values are more spread out.

Population Variance
01 calculate population variance
02 Average population variance

Important Results related to
Variance

Discrete distribution
BERNOULLI 01 BINOMIAL 02 UNIFORM 03
POISSON 04 GEOMETRIC 05
Click here to add text
Your
title
here

History
Bernoulli Distribution
This distribution, named after the Swiss
mathematician Jacob Bernoulli, describes a
discrete random variable that can only take
two possible outcomes: it equals 1 with a
probability of p, or 0 with a probability of 1 minus p.

Simplest one = Bernoulli
Random variable
1. What Is It?
 A Bernoulli distribution models a single “yes/no” trial with two possible outcomes:
 Success (1) with probability p
 Failure (0) with probability 1 –p
 2. Why It Matters
 Simplest building block for more complex models (e.g., Binomial).
 Binary decisions: pass/fail, on/off, defect/no defect.
 Foundation for machine learning (logistic regression), A/B testing, reliability studies.
 where p is probability of success

(Assumptions & Example)
Example:
A student takes a surprise quiz. The outcome is either:
1 (Success): The student passes the quiz.
0 (Failure): The student fails the quiz.
If the probability of passing is p=0.7, then this is a Bernoulli random variable with:
P(X=1)=0.7
P(X=0)=0.3
Assumption Why It Matters
Only Two Outcomes Models strictly binary events (e.g., spam/not spam).
Constant Probability Ensures p is fixed for the trial’s validity.
Mutually Exclusive Success and failure cannot occur together.
Single Trial One event per distribution (repeat → Binomial).
Independence (when reused) Trials don’t influence each other (Bernoulli process).

Graphical
Representation(p=0.7,x=1)

Binomial Random Variable
When to Use
• Clinical Trial Outcomes: Counting how many of 100 patients respond positively to a new
drug, assuming each patient has the same response probability.
• Email Campaign: Among 1,000 recipients, tallying how many open the email when each
has a fixed open rate.
• Manufacturing Defects: Inspecting 200 widgets and counting the defective ones when
each widget has the same defect probability.

Example And Assumptions
Fixed Number of Trials Ensures exactly n opportunities for success.
ndependent Trials One trial’s outcome doesn’t influence another’s.
Constant Probability Success probability p is the same across trials.
Binary Outcome per Trial Each trial yields only success or failure.
Find probability of Passing exactly 3 quizzes?

Expectation (Mean) and
Variance(dispersion)

Graphical Form( n=15 and
p=0.8)

Graphical Form( n=5 ,p=0.2
and x=5)

History
 The concept of uniform distribution, in the context of
probability, is not attributed to a single discoverer. While the
term "uniform distribution" was defined later, the idea of equal
probability for all outcomes within a certain range has roots in
earlier works. Specifically, the continuous uniform distribution
can be traced back to Thomas Bayes' work on conditional
probabilities in 1763

Uniform Random Variable
1.What it is?
All outcomes within a specified range (continuous) or set (discrete) are equally likely.
2. When to Use
• Lottery Draw for Limited Prizes: Randomly picking 20 winners out of 2,000 lottery entries—each entry
has exactly a 1/2000 chance.
• Random Student Selection: Choosing one student out of a class of 30 for a presentation slot, ensuring
every student is equally likely.

(Example and Assumption)
Equal Likelihood Ensures fairness: every outcome has probability 1/m.
Fixed Set of Values Outcomes must be predetermined (e.g., ticket IDs 1–1,000).
Independence
One draw does not affect the next (if sampling with
replacement).

Graphical
form(a=5,b=15,x=10)
PB
RV

History
The Poisson distribution was discovered
by Siméon Denis Poisson, a French
mathematician and physicist, in 1837.
He developed this probability
distribution while studying the number
of rare events occurring within a fixed
interval of time or space, according to
a publication from ResearchGate.

Poisson Distribution
1.What it is?
Models the count of events occurring in a fixed interval of time or space, given a constant average
rate λ and independent occurrences.
2. When to Use
Emergency Room Arrivals: Counting how many patients arrive in one hour at a hospital in ER.
Web Server Traffic: Number of page requests received by a website per minute.
Manufacturing Defects on a Conveyor: Number of flaws detected per meter of produced fabric.
where λ: average rate of events per interval (mean and variance both = λ)

Example and Assumption
A small clinic receives an average of 6 patients per hour in the emergency room.
You're interested in finding the probability of receiving exactly 8 patients in an hour.
Independent Events One event’s occurrence does not impact another’s.
Constant Rate λ Average events per interval remain stable.
Events Are Rare Probability of two or more simultaneous events ≈ 0.
Non overlapping Intervals
‐ Counts in separate intervals are independent.

Graphical form(x=8, =‫גּ‬
6 )
PB
RV

History(GEOMETRIC
DISTRIBUTION)
The geometric distribution was not
discovered by one person but emerged from
the study of Bernoulli trials—counting the
number of trials until the first success. It
evolved through the foundational work of
Jacob Bernoulli and others in early probability
theory.

Geometric Random Variable
Why to use?
A geometric distribution tells you how many tries you’ll need to get your first “win” when each
attempt has the same chance of succeeding and attempts don’t affect each other.
Why It’s Useful
Planning: Estimate how long—or how many attempts—you’ll likely need.
Resource Allocation: Budget time or effort based on expected attempts (e.g.,
call center staffing).
Risk Assessment: Understand the “worst-case” tries you might face before
success.

MEMORYLESSNESS PROPERTY
Definition:
The Geometric distribution is memoryless, meaning the probability of success in future trials
doesn’t depend on past failures.
.
Why It’s Called “Memoryless”
Because the process "forgets" how many failures have already happened — only future
matters.
✅ Only Two Memoryless Distributions
•Geometric distribution (discrete)
•Exponential distribution (continuous)

🧪 When to Use
•Repeated independent trials
•Each trial has a constant success probability p
•You're counting how many trials until the first success
📌 Example
A call center agent keeps picking up calls until they get a difficult customer
(considered a "success"). Each call is independent, and the chance of a
difficult customer is constant, say 10%.
If the agent has already answered 5 easy calls, the probability that the next
call is a difficult one is still 10% — unchanged by the past.
This reflects the memoryless property of the geometric distribution.

Notations
•x: Represents the total number of trials needed to
achieve (k) successes in a negative binomial setting.
•k: Denotes the count of successful outcomes targeted in
the experiment.
•P: The likelihood of success occurring on any single trial.
•q: The chance that a trial results in failure.

SOLUTION
Hint for (c):
Use the complement rule

Example
STATEMENT
A batch of 20 vaccine doses contains 12 effective doses and 8 defective ones. Doses are tested one at a
time without replacement until the 3rd effective dose is found.
What is the probability that this 3rd effective dose appears exactly on the 5th test?

CONTINEOUS DISTRIBUTION
UNIFORM
CONTINEOUS 01
EXPONENTIAL
DISTRIBUTION 02
NORMAL
DISTRIBUTION
03
Click here to add text Click here to add text Click here to add text

Probability Density Function
(PDF)
Definition
 Imagine you're trying to describe the likelihood of a continuous event happening, like someone's
height, the temperature tomorrow, or the amount of rain in a day. For these continuous things, you
can't say "what's the probability that someone is exactly 170.00000... cm tall?" because the
probability of any single exact value is essentially zero (there are infinitely many possibilities).
 Instead, a Probability Density Function (PDF) tells you where values are more "dense" or
concentrated. It's like a curve on a graph, and the higher the curve, the more likely it is for the
value to fall within a small range around that point. The area under the curve between two points
tells you the probability that the event falls within that range.
A Probability Density Function (PDF) is a function that describes the likelihood of a continuous random
variable I taking on a particular value.

PDF(Probability Density
FUNCTION)

Example: Human Height
Let's consider the height of adult males in a particular country.
•What it is: The PDF for human height would be a bell-shaped curve (often
approximated by a normal distribution).
•How to read it: If the curve is highest around 175 cm, it means that heights
close to 175 cm are the most common. The curve would be lower at 150
cm or 200 cm, indicating those heights are less common.
•Example Question: What is the probability that a randomly selected adult
male is between 170 cm and 180 cm tall?
Answer
Approach: You would calculate the area under the PDF curve between
170 and 180 cm. This area represents the probability. You wouldn't ask
"what's the probability that someone is exactly 175 cm tall?" because for a
continuous variable, that probability is infinitesimally small (approaching
zero).

Joint Probability Density Function (Joint PDF)
Sometimes, you're interested in the likelihood of two or more continuous
events happening together. For example, what's the likelihood of someone
having a certain height and a certain weight simultaneously?
A Joint Probability Density Function (Joint PDF) is like a 3D landscape that
shows you where pairs (or more) of values are most likely to occur together. I.
The volume under this surface over a specific region gives you the probability
that both events fall within their respective ranges.

Conditional Probability Density Function (Conditional
PDF)
A Conditional PDF answers the question: "Given that we already know
something about one variable, what does that tell us about the probability
distribution of another variable?"
Think about it like this: If you know someone is very tall, what does that tell
you about their likely weight? It's probably more likely that they are heavier
than average, even though there's still a range of possibilities. The
Conditional PDF updates our understanding of one variable's likelihood
based on new information about another.

Cumulative Distribution
Function(CDF)

Joint CDF( Cumulative
Distribution Function)

CONTINEOUS UNIFORM
DISTRIBUTION
01

Contineous Uniform
Distribution
Definition
A continuous random variable follows a distribution if all
outcomes in the interval are equally likely.
Assumptions

Mean, Variance and Standard deviation

example
 Problem Statement:
 A pharmaceutical company stores vaccine doses in standardized cold storage units. The
‑
temperature (in degrees Celsius) inside each unit is monitored continuously and is known to
vary uniformly between -2and 2.Find Probability it vary between -1 and 1.

what is Exponential
Distribution
 Exponential Distribution
The exponential distribution is used to model how long you have
to wait for something to happen if it happens randomly and
continuously over time.
 Think of Real-Life Examples:
1. How long you wait at a bus stop for the next bus.
2. Time between phone calls at a call center.
3. Time until a machine breaks down.
4. If these events happen at a constant average rate, then the
time you wait between them follows an exponential distribution.

For small‫גּ‬ and large ‫גּ‬

Similarity of Exponential and
Geometric

similarity of exponential and
poisson

• The normal distribution, also known as the Gaussian distribution, was first
described by Abraham de Moivre in 1733, and later rediscovered and further
developed by Carl Friedrich Gauss in the early 19th century. Gauss is
often more closely associated with it due to his extensive work and application
of the distribution, particularly in the context of measurement errors in
astronomy.
HISTORY

NORMAL DISTRIBUTION
 What is it?
 The Normal distribution is a bell-shaped curve that shows how
values of a random variable are spread around the mean
(average).
 It is one of the most important and widely used distributions in
statistics.

Standard Normal Distribution
(MEAN =0 , VARIANCE =1)

Central Limit Theoram
UNIFORM DISTRIBUTION
EXPONENTIAL DISTRIBUTION

Applications & Further Reading
Applications in statistics, engineering,
ML, finance
Recommended textbooks and online
resources

Guide to Probability & Distributions: From Fundamentals to Advanced Concepts

More Related Content

Similar to Guide to Probability & Distributions: From Fundamentals to Advanced Concepts (20)

Recently uploaded (20)

Guide to Probability & Distributions: From Fundamentals to Advanced Concepts