Section 6 - Chapter 3 - Introduction to Probablity

Chapter 3 - Introduction to Probability
Section 6 – Statistics Analysis
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Agenda
 Introduction to Probability
 The Search for the High-Probability Trade
 Properties of Probability
 The Probability Distribution

Probability in Statistics
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Key Facts:
1. Basics of Probability
• Definition: Probability measures the likelihood of an event occurring, ranging between 0 (impossible)
and 1 (certain).
• Formula:
Types of Events:
o Independent Events: The occurrence of one event does not affect the other.
Example: Rolling two dice.
o Dependent Events: The occurrence of one event affects the other.
Example: Drawing cards from a deck without replacement.
o Mutually Exclusive Events: Two events cannot happen at the same time.
Example: Getting heads or tails on a coin flip.
o Non-Mutually Exclusive Events: Events that can happen together.
Example: Drawing a red card and a face card in a deck.

Probability Distributions
Distribution Description Example
Uniform All outcomes are equally likely Rolling a fair die
Bernoulli Single trial with success/failure
Tossing a coin (heads = 1, tails =
0)
Binomial
Number of successes in nnn
trials
Flipping a coin 10 times
Poisson
Number of occurrences in a
fixed interval
Number of customer arrivals in 1
hour
Normal (Gaussian)
Bell-shaped, symmetric around
mean
Heights of people, IQ scores
Exponential Time until an event occurs Time between arrivals of buses

Common Misconceptions
🚫 Misconception: If an event hasn’t happened in a while, it’s "due" to occur.
✅ Reality: Each independent trial (like a fair coin flip) has the same probability.
🚫 Misconception: A high probability means certainty.
✅ Reality: Even high-probability events can fail to happen.
🚫 Misconception: If P(A B)P(A | B)P(A B) is high, then P(B A)P(B | A)P(B A) must
∣ ∣ ∣ ∣
also be high.
✅ Reality: Not necessarily, as seen in Bayes' Theorem.

Normal Distribution
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Normal Distribution
The Normal Distribution, also known as the Gaussian Distribution, is one of the most important
probability distributions in statistics. It models many real-world phenomena such as heights, test
scores, IQ levels, and measurement errors.
1. Characteristics of Normal Distribution
✅ Bell-shaped curve: Symmetrical around the mean.
✅ Mean = Median = Mode: The highest point is at the mean μmuμ.
✅ Defined by two parameters:
• μmuμ (mean): Center of the distribution.
• sigmaσ (standard deviation): Controls the spread.
✅ Total area under the curve = 1
✅ Follows the empirical rule (68-95-99.7 rule) (see below).

Normal Distribution
2. Probability Density Function (PDF)
The normal distribution is given by the formula:
where:
• x = random variable
• μ = mean
• σ = standard deviation
• e = Euler’s number (≈2.718)
• πpiπ = Pi (≈3.1416)

Normal Distribution
3. Standard Normal Distribution (Z-Score)
A standard normal distribution is a normal distribution with:
• Mean μ=0mu = 0μ=0
• Standard deviation σ=1sigma = 1σ=1
To convert any normal variable X to standard normal form:
where Z is called the Z-score, representing how many standard deviations XXX is from
the mean.
🔹 Example: If a student scores 85 on a test where μ=70 , σ=10
Z=(85−70
) /10 =1.5This means the student is 1.5 standard deviations above the
mean.

Normal Distribution
4. Empirical Rule (68-95-99.7 Rule)
In a normal distribution:
68% of values fall within 1 standard deviation ( ± σ)
𝜇
95% of values fall within 2 standard deviations ( ± 2 )
𝜇 𝜎
99.7% of values fall within 3 standard deviations ( ± 3 )
𝜇 𝜎
📊 Example: If human IQ follows a normal distribution with =100 and =15
𝜇 𝜎
68% of people have IQs between 85 and 115.
95% of people have IQs between 70 and 130.
99.7% of people have IQs between 55 and 145.

Normal Distribution
5. Applications of Normal Distribution
🔹 Standardized Testing: SAT, IQ scores follow normal distribution.
🔹 Measurement Errors: Errors in scientific measurements tend to be normally distributed.
🔹 Stock Market Returns: Approximate a normal distribution in short periods.
🔹 Quality Control: Used in manufacturing defect analysis.
6. Normal vs. Other Distributions
Feature Normal Binomial Poisson Exponential
Type Continuous Discrete Discrete Continuous
Shape Bell-shaped Skewed (for small p) Skewed (for small λ) Right-skewed
Parameters μ,σ n,p λ λ
Example Heights, IQ Coin flips Calls per hour Time between arrivals

Normal Distribution
7. Central Limit Theorem (CLT)
The Central Limit Theorem (CLT) states that the sum (or mean) of a large number of
independent random variables, regardless of their original distribution, will
approximate a normal distribution.
✅ Even if data is not normally distributed, its sample mean will be!
✅ Works well for sample sizes n>30n > 30n>30.
🔹 Example: If we repeatedly take samples of 50 students’ test scores, their average
test score will follow a normal distribution, even if individual test scores don’t.

Normal Distribution
8. Finding Probabilities Using Z-Tables
To find probabilities, use a Z-table, which gives cumulative probabilities
for standard normal distribution.
📌 Example: Find P(X>85) where μ=70 , σ=10.
1. Convert to Z-score: Z=(85−70)/10=1.5
2. From the Z-table, P(Z<1.5) = 0.9332
3. Since we need P(X>85) =1−0.9332=0.0668
So, 6.68% of scores are above 85.

Skewness & Kurtosis
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Skewness & Kurtosis
Skewness and Kurtosis are statistical measures that describe the shape of a probability distribution
compared to a normal distribution.
1. Skewness: Measuring Symmetry
Definition: Skewness measures how asymmetric a distribution is around its mean.
Types of Skewness
🔹 Symmetric Distribution (Skewness = 0)
• Mean = Median = Mode
• Example: Normal distribution
🔹 Positive Skew (Right-Skewed, Skewness > 0)
• Tail extends to the right
• Mean > Median > Mode
• Example: Income distribution, waiting times

Skewness & Kurtosis
🔹 Negative Skew (Left-Skewed, Skewness < 0)
• Tail extends to the left
• Mean < Median < Mode
• Example: Test scores with many high scores
Skewness Value Interpretation
= 0 Symmetrical
>0 Right-skewed
<0 Left-skewed
>1 or < -1 Highly skewed
Interpretation of Skewness Values
Formula:

Kurtosis
Kurtosis: Measuring Tailedness
Definition: Kurtosis measures whether a distribution has heavy or light tails compared to a normal distribution.
Types of Kurtosis
🔹 Meso kurtic (Kurtosis = 3)
• Normal distribution
• Moderate tails
🔹 Leptokurtic (Kurtosis > 3)
• Heavy tails, many extreme values
• Example: Stock market crashes
🔹 Platy kurtic (Kurtosis < 3)
• Light tails, few extreme values
• Example: Uniform distribution

Kurtosis
Types of Kurtosis
1. Mesokurtic (Kurtosis = 3)
✅ Definition: The distribution has a kurtosis of 3, meaning it resembles the
normal distribution.
✅ Characteristics:
• Moderate tails (neither too heavy nor too light).
• Similar peak height to a normal distribution.
✅ Example: Normal Distribution, Exponential Distribution (under some
conditions).

Kurtosis
Types of Kurtosis
2. Leptokurtic (Kurtosis > 3)
✅ Definition: The distribution has heavy tails, meaning more extreme values
(outliers) than a normal distribution.
• Higher peak and fatter tails.
• More frequent extreme values.
✅ Example: Stock market crashes, financial returns, insurance claims,
earthquake magnitudes.

Kurtosis
Types of Kurtosis
3. Platykurtic (Kurtosis < 3)
✅ Definition: The distribution has light tails, meaning fewer extreme values
than a normal distribution.
• Lower peak and thinner tails.
• Less variation and fewer outliers.
✅ Example: Uniform distribution, Rolling a fair die, Student test scores with
minimal variance.

Kurtosis
Interpretation of Kurtosis Values
Kurtosis Value Interpretation
= 3 Normal distribution (Mesokurtic)
>3 Heavy tails (Leptokurtic)
<3 Light tails (Platykurtic)
Formula of Kurtosis

Kurtosis
Comparison: Skewness vs. Kurtosis
Feature Skewness Kurtosis
Definition Measures asymmetry Measures tail heaviness
Focus Left or right tail behavior Extreme values (outliers)
Key Values >0 (right), <0 (left), =0(symmetric)
>3(leptokurtic), <3(platykurtic), =3
(normal)
Example Income distribution (right-skewed) Stock returns (leptokurtic)
Real-World Applications
🔹 Finance: Skewness & Kurtosis help assess risk in stock returns.
🔹 Quality Control: Detects deviations from normal performance.
🔹 Economics: Identifies income inequalities.
🔹 Social Sciences: Measures biases in survey responses.

Chapter 1 - Behavioral Finance
Next Section 7 - Behavioural Finance
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Section 6 - Chapter 3 - Introduction to Probablity

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