Mathematical Statistics

🎯 Learning Objectives

By the end of this lecture, you will be able to:

Identify and verify the five properties of binomial experiments in financial and economic contexts
Apply the binomial probability formula to calculate probabilities for investment, credit risk, and market scenarios
Interpret binomial probability distributions and their parameters in economic applications
Compute mean and variance of binomial random variables for financial modeling
Use maximum likelihood estimation to estimate success probabilities from sample data
Apply binomial models to real-world financial and economic problems

📋 Overview

📚 Topics Covered Today

Binomial Probability Distribution - Definition and mathematical properties
Binomial Experiments - Identifying conditions in finance and economics
Probability Calculations - Using formulas and statistical tables
Mean and Variance - Expected value and risk measures
Applications - Portfolio management, credit risk, market analysis
Maximum Likelihood Estimation - Parameter estimation from data

📖 Definition: Binomial Experiment

📝 Definition 1: Binomial Experiment

A binomial experiment possesses the following properties:

🔢 The experiment consists of a fixed number, \(n\), of identical trials
⚖️ Each trial results in one of two outcomes: success (\(S\)) or failure (\(F\))
🎯 The probability of success \(p\) remains constant from trial to trial, with \(q = (1 - p)\)
🔗 The trials are independent of each other
📊 The random variable \(Y\) counts the number of successes in \(n\) trials

🤔 Think-Pair-Share Activity

💭 Activity 1: Identifying Binomial Experiments (3 minutes)

🧠 Think (1 minute): Consider the following financial scenario individually:

“A credit analyst reviews 10 loan applications from different borrowers. Each application is either approved or rejected based on credit scores. Historical data shows a 65% approval rate.”

👥 Pair (1 minute): Discuss with your neighbor - does this meet all five binomial experiment properties?

03:00

🗣️ Share (1 minute): Groups share their conclusions with the class

📌 Example 1: Credit Risk Management

💳 Example 1

A bank’s loan portfolio contains four independent corporate loans, each with a historical 5% probability of default over the next year. The risk manager is interested in \(Y\), the number of loans that will default. Is this a binomial experiment?

✅ Solution

Let’s verify each property:

✅ Fixed trials: Four independent loans (\(n = 4\))
✅ Two outcomes:
- Success (\(S\)): loan defaults
- Failure (\(F\)): loan does not default
✅ Constant probability: \(p = P(S) = 0.05\)

✅ Independence: Loans are from different sectors/independent
✅ Count successes: \(Y\) = number of defaults

Conclusion: Yes, this is a binomial experiment! The bank can use binomial distribution to assess portfolio risk.

📌 Example 2: Market Survey

📊 Example 2

An economist surveys 10 randomly selected consumers from a large population where 40% prefer brand A over competitors. Let \(Y\) be the number who prefer brand A. Does this meet binomial requirements?

💡 Important Remark

When sampling without replacement from a finite population, trials are technically dependent. However:

Rule of Thumb: If the sample size is less than 5% of the population (\(n < 0.05N\)), the dependency is negligible and we can approximate with the binomial distribution.

For large consumer markets, this condition is easily satisfied!

📝 Quiz #1: Identifying Binomial Experiments

Which of the following is NOT a binomial experiment?

Monitoring 20 stock trades where each results in profit or loss
Surveying investors until finding 5 who own cryptocurrency
Testing 15 products from a production line for defects
Checking 30 loan applications for completeness

02:00

🧮 The Binomial Distribution

📝 Definition 2: Binomial Distribution

A random variable \(Y\) has a binomial distribution based on \(n\) trials with success probability \(p\) if and only if:

\[ p(y) = \binom{n}{y}p^yq^{n-y}, \quad y=0,1,2,\dotsc, n \quad \text{and} \quad 0\le p \le 1 \]

where \(q = 1 - p\) and \(\binom{n}{y} = \frac{n!}{y!(n-y)!}\)

Notation: We write \(Y \sim \text{Binomial}(n, p)\) or \(Y \sim B(n, p)\)

💡 Binomial Theorem

\[ (q+p)^n = \binom{n}{0}q^n+\binom{n}{1}pq^{n-1}+\binom{n}{2}p^2q^{n-2}+\cdots+\binom{n}{n}p^n \]

Since \(p + q = 1\), we have \((q+p)^n = 1\), confirming that \(\sum_{y=0}^{n}p(y) = 1\) ✓

🔍 Interactive Exploration

🌐 Explore Binomial Distributions

Visit this interactive applet: Binomial Distribution Applet

Activity: Experiment with different parameter values:

📊 Set \(n = 10\), \(p = 0.5\) → Symmetric distribution (like a fair coin)
📉 Set \(n = 10\), \(p = 0.1\) → Right-skewed (rare events)
📈 Set \(n = 10\), \(p = 0.9\) → Left-skewed (common events)

Financial Interpretation: Observe how changing default probability \(p\) affects the distribution shape!

🤔 Think-Pair-Share Activity #2

🧩 Activity 2: Understanding the Formula Components (4 minutes)

🧠 Think (1.5 min): Why do we multiply \(\binom{n}{y}\), \(p^y\), and \(q^{n-y}\) together? What does each component represent in a portfolio of risky assets?

👥 Pair (1.5 min): Explain your reasoning to your partner. - What is \(\binom{n}{y}\)? (Number of ways to select which assets default) - What is \(p^y\)? (Probability those specific assets default) - What is \(q^{n-y}\)? (Probability remaining assets don’t default)

04:00

🗣️ Share (1 min): Volunteer pairs share insights with the class

📌 Example 3: Investment Portfolio

💼 Example 3

A venture capital fund invests in 5 startup companies. Historical data shows that 5% of startups fail within the first year. Find the probability that at least one startup fails.

🔢 Solution

Let \(Y\) = number of failed startups. Then \(Y \sim \text{Binomial}(n=5, p=0.05)\).

\[ P(\text{at least one fails}) = 1 - P(Y = 0) \]

\[ P(Y = 0) = \binom{5}{0}(0.05)^0(0.95)^5 = 1 \times 1 \times (0.95)^5 = 0.774 \]

Therefore: \[ P(Y \geq 1) = 1 - 0.774 = 0.226 \]

Interpretation: There’s a 22.6% chance that at least one startup will fail, despite the individual failure rate being only 5%. This demonstrates the importance of diversification!

📌 Example 4: Pharmaceutical Investment

💊 Example 4

A pharmaceutical company knows that 30% of experimental drugs pass clinical trials without their new technology. They claim a new screening method increases success. With the new method, 9 out of 10 drugs passed trials. If the method were worthless, what’s \(P(Y \geq 9)\)?

📊 Solution Setup

Let \(Y\) = number of drugs passing trials. If method is worthless: \(p = 0.3\), \(n = 10\).

We need to find \(P(Y \geq 9) = P(Y = 9) + P(Y = 10)\)

📌 Example 4: Solution

Probability of exactly nine successes:

\[ P(Y = 9) = \binom{10}{9} (0.3)^9 (0.7)^1 = 10 \times (0.3)^9 \times 0.7 = 0.000138 \]

Probability of exactly ten successes:

\[ P(Y = 10) = \binom{10}{10} (0.3)^{10} (0.7)^0 = (0.3)^{10} = 0.000006 \]

Therefore: \[ P(Y \geq 9) = 0.000138 + 0.000006 = 0.000144 \]

🎯 Statistical Interpretation

If the screening method is ineffective, observing 9+ successes has probability 0.0144% (about 1 in 7,000!). This provides very strong evidence that the new screening method is effective. A pharmaceutical company would likely invest in this technology based on such results.

📝 Quiz #2: Calculating Probabilities

An investor makes 6 independent trades with 50% probability of profit on each. What is the probability of exactly 4 profitable trades?

Hint: Use \(n=6\), \(p=0.5\), \(y=4\) in the binomial formula.

0.234
0.313
0.156
0.500

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📌 Example 5: Using Statistical Tables

📋 Example 5

A bond portfolio contains bonds with 5% default probability. If the fund holds 20 bonds, find the probability that at least four bonds default.

🔢 Solution

With \(Y \sim \text{Binomial}(n=20, p=0.05)\):

\[ P(Y\geq 4) = 1 - P(Y\leq 3) = 1 - \sum_{y=0}^3p(y) \]

We can calculate each term or use cumulative binomial tables. Using statistical software or tables:

\(P(Y \leq 3) \approx 0.9841\)

Therefore: \[ P(Y \geq 4) = 1 - 0.9841 = 0.0159 \]

Risk Assessment: Only about 1.6% chance of having 4+ defaults, suggesting the portfolio is relatively safe under current assumptions.

📊 Using Binomial Tables

📚 Cumulative Binomial Tables

Tabulated values represent cumulative probabilities: \[ P(Y \leq a) = \sum_{y=0}^{a}p(y) \]

Steps for \(n=20\), \(p=0.05\), finding \(P(Y \leq 3)\):

➊ Locate the section for \(n=20\)
➋ Find the column for \(p=0.05\)
➌ Read the cumulative probability at row \(y=3\)

Modern Approach: Use statistical software for exact calculations:

R: pbinom(3, 20, 0.05) gives \(P(Y \leq 3)\)
Python: scipy.stats.binom.cdf(3, 20, 0.05)
Excel: =BINOM.DIST(3, 20, 0.05, TRUE)

🎯 Key Theorem

📝 Theorem 1: Mean and Variance

Let \(Y\) be a binomial random variable based on \(n\) trials with success probability \(p\). Then:

\[ \mu = E(Y) = np \]

\[ \sigma^2 = V(Y) = npq = np(1-p) \]

\[ \sigma = SD(Y) = \sqrt{npq} \]

where \(q = 1 - p\).

💡 Financial Interpretation

📊 Expected value \(E(Y) = np\): Expected number of defaults, claims, or successes
📉 Variance \(V(Y) = npq\): Measure of risk/uncertainty; maximum when \(p = 0.5\)
📐 Standard deviation \(\sigma\): Typical deviation from expected value (used in risk management)

🤔 Group Activity: Portfolio Risk

🏦 Activity 3: Understanding Expected Value (5 minutes)

Scenario: A bank’s credit card portfolio has 1000 accounts, each with 2% monthly default probability.

Tasks (in groups of 3-4):

1. Calculate expected number of defaults per month (1 min)

2. Calculate the standard deviation (1 min)

3. If portfolio grows to 10,000 accounts, what happens to expected defaults and standard deviation? Does the default probability change? (2 min)

05:00

4. Present your insights and discuss risk implications (1 min/group)

📌 Example 6: Market Research

📊 Example 6

A market research firm surveys 20 randomly selected consumers about a new product. 6 respondents expressed interest in purchasing. Estimate \(p\), the true proportion of the population interested in the product.

🔍 Solution: Maximum Likelihood Approach

Let \(Y\) = number interested in product (observed \(Y = 6\))
\(Y \sim \text{Binomial}(n=20, p=?)\)
Likelihood of observing \(Y=6\):

\[ L(p) = P(Y = 6 | p) = \binom{20}{6} p^6 (1 - p)^{14} \]

Question: What value of \(p\) maximizes this likelihood?

🎓 Maximum Likelihood Estimation

To maximize \(L(p) = \binom{20}{6} p^6 (1 - p)^{14}\):

🎯 \(\binom{20}{6}\) is constant relative to \(p\)
📈 Since \(\ln(x)\) is strictly increasing, maximize log-likelihood instead:

\[ \ln L(p) = \ln\binom{20}{6} + 6\ln(p) + 14\ln(1-p) \]

Differentiate with respect to \(p\):

\[ \frac{d\ln L}{dp} = \frac{6}{p} - \frac{14}{1-p} \]

🔍 Finding the Maximum Likelihood Estimate

Set the derivative equal to zero:

\[ \frac{6}{p} - \frac{14}{1-p} = 0 \]

Multiply both sides by \(p(1-p)\):

\[ 6(1-p) - 14p = 0 \]

\[ 6 - 6p - 14p = 0 \]

\[ 6 = 20p \]

\[ p = \frac{6}{20} = 0.3 \]

✅ Solution Verification

✓ Final Solution

Check the second derivative: \[ \frac{d^2\ln L}{dp^2} = -\frac{6}{p^2} - \frac{14}{(1-p)^2} \]

This is always negative for \(0 < p < 1\), confirming a maximum at \(p = 6/20\).

Maximum Likelihood Estimate: \[ \boxed{\hat{p} = \frac{6}{20} = 0.30 = 30\%} \]

🎯 General Result

For binomial data, the Maximum Likelihood Estimator (MLE) is always: \[ \hat{p} = \frac{Y}{n} = \frac{\text{number of successes}}{\text{number of trials}} \]

This is simply the sample proportion - intuitive and optimal!

📝 Quiz #3: Mean and Variance

A loan portfolio has 25 loans, each with 40% default probability (independent). What are \(E(Y)\) and \(SD(Y)\)?

Hint: \(E(Y) = np\) and \(SD(Y) = \sqrt{npq}\)

\(E(Y) = 10\), \(SD(Y) = 2.45\)
\(E(Y) = 12\), \(SD(Y) = 3.00\)
\(E(Y) = 10\), \(SD(Y) = 6.00\)
\(E(Y) = 15\), \(SD(Y) = 2.45\)

03:00

🤔 Think-Pair-Share Activity #3

🏭 Activity 4: Quality Control Application (5 minutes)

Scenario: A manufacturing company’s historical defect rate is 2%. Today’s inspection found 5 defects in a sample of 50 products.

🧠 Think (2 min): Is this evidence that the defect rate has increased? Under \(p = 0.02\), what’s \(E(Y)\)?

👥 Pair (2 min): Calculate (or estimate) \(P(Y \geq 5)\) when \(n=50\), \(p=0.02\). Use the fact that \(E(Y) = 1\). Is 5 unusually high?

05:00

🗣️ Share (1 min): Present conclusions about whether quality has deteriorated

📝 Summary

✅ Key Takeaways

Five Properties of Binomial Experiments:

🔢 Fixed number of trials \(n\)
⚖️ Two outcomes: success or failure
🎯 Constant probability \(p\)
🔗 Independent trials
📊 Count of successes \(Y\)

Essential Formulas: \[ p(y) = \binom{n}{y}p^yq^{n-y}, \quad E(Y) = np, \quad V(Y) = npq, \quad \hat{p}_{\text{MLE}} = \frac{Y}{n} \]

Applications: Credit risk, portfolio management, quality control, market research, insurance claims

📚 Practice Problems

📝 Homework Problems

💻 Investment Portfolio: A venture capital fund invests in 10 startups. If 5% typically fail within one year, find:
- \(P(\text{at most 2 fail})\)
- Expected failures and standard deviation
- Is observing 3 failures unusual?
🏥 Insurance Claims: An insurance company expects 8% of policyholders to file claims. For 15 randomly selected policies:
- Find \(P(\text{exactly 2 claims})\)
- Calculate \(E(Y)\) and \(SD(Y)\)
- If 4 file claims, should the company investigate?
📱 Market Research: Historical data shows 35% of customers convert after seeing an ad. Sample of 20:
- Find \(P(Y = 7)\)
- Find \(P(Y \geq 10)\)
- If 12 converted, estimate the new conversion rate using MLE

Due: Next class meeting

🌐 Additional Resources

📖 For Further Study

Interactive Tools:

🔗 Binomial Distribution Calculator
🎓 Khan Academy: Binomial Probability
📺 StatQuest YouTube: Binomial Distribution

Statistical Software:

📊 R: dbinom(), pbinom(), qbinom(), rbinom()
🐍 Python: scipy.stats.binom
📈 Excel: =BINOM.DIST()

Textbook Reading:

Chapter 3.4: The Binomial Distribution
Chapter 3.5: Applications and Examples
Review: Expected Value and Variance (Chapter 3.2)

👋 Thank You!

📬 Contact Information:

Samir Orujov
Assistant Professor
School of Business ADA University

📧 Email: sorujov@ada.edu.az
🏢 Office: D312 ⏰ Office Hours: By appointment

📅 Next Class:

Topic: Poisson Distribution
Reading: Chapter 3.6
Preparation: Review binomial concepts

⏰ Reminders:

✅ Complete all homework problems
✅ Review MLE concept thoroughly
✅ Work hard

❓ Questions?

💬 Open Discussion (5 minutes)

Feel free to ask questions about:

📊 Binomial distribution properties and assumptions
🧮 Calculating probabilities using formulas or tables
📈 Mean and variance interpretations
🎯 Maximum likelihood estimation methodology
💼 Applications to finance, economics, or your research

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