Section 6 - Chapter 3 - Introduction to Probablity

Section 6 - Chapter 3 - Introduction to Probablity - Presented by Rohan Sharma - The CMT Coach - Chartered Market Technician CMT Level 1 Study Material - CMT Level 1 Chapter Wise Short Notes - CMT Level 1 Course Content - CMT Level 1 2025 Exam Syllabus Visit Site : www.learn.ptaindia.com and www.ptaindia.com

1. Chapter 3 - Introduction to Probability Section 6 – Statistics Analysis Presented By : This Content is Copyright Reserved Rights Copyright 2025@PTAIndia

2. Agenda Introduction to Probability The Search for the High-Probability Trade Properties of Probability The Probability Distribution This Content is Copyright Reserved Rights Copyright 2025@PTAIndia

3. Probability in Statistics Presented By : This Content is Copyright Reserved Rights Copyright 2025@PTAIndia

4. Probability in Statistics 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.

5. Probability in Statistics 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

6. Probability in Statistics 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.

7. Normal Distribution Presented By : This Content is Copyright Reserved Rights Copyright 2025@PTAIndia

8. 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 μ\ muμ . ✅ Defined by two parameters: • μ\ muμ (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 ).

9. 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) • π\ pi π = Pi (≈3.1416)

10. Normal Distribution 3. Standard Normal Distribution (Z-Score) A standard normal distribution is a normal distribution with: • Mean μ=0\ mu = 0 μ=0 • Standard deviation σ=1\ sigma = 1 σ=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 .

11. Normal Distribution 4. Empirical Rule (68-95-99.7 Rule ) In a normal distribution : 68 % of values fall within 1 standard deviation ( 𝜇 ± σ) 95 % of values fall within 2 standard deviations ( 𝜇 ± 2𝜎) 99.7 % of values fall within 3 standard deviations ( 𝜇 ± 3𝜎) 📊 Example: If human IQ follows a normal distribution with 𝜇= 100 and 𝜎= 15 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.

12. 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

13. 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 .

14. 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 From the Z-table, P(Z<1.5 ) = 0.9332 Since we need P(X>85) = 1− 0.9332=0.0668 So, 6.68% of scores are above 85 .

15. Skewness & Kurtosis Presented By : This Content is Copyright Reserved Rights Copyright 2025@PTAIndia

16. 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

17. 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 = Symmetrical > Right-skewed < Left-skewed > 1 or < - 1 Highly skewed Interpretation of Skewness Values Formula:

18. Skewness & Kurtosis

19. 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

20. Kurtosis

21. 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 ).

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

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

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

25. 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 > (right), < (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.

26. Chapter 1 - Behavioral Finance Next Section 7 - Behavioural Finance Presented By : This Content is Copyright Reserved Rights Copyright 2025@PTAIndia