Chief AI Officer Program: ML Essentials

Machine Learning: From Probability to Prediction

Author

Affiliation

Vadim Sokolov

George Mason University

Probability Distributions

Why Distributions Matter

Machine learning is built on probability
Distributions describe uncertainty in data
Three fundamental distributions:
- Binomial: Binary outcomes (yes/no, win/lose)
- Poisson: Count data (arrivals, events)
- Normal: Continuous measurements (heights, returns)

Why Should Executives Care?

Business Question	Distribution
Will the customer buy?	Binomial
How many orders today?	Poisson
What’s the forecast error?	Normal

Choosing the right distribution is the first step in building a reliable model. Wrong distribution = wrong predictions!

Binomial Distribution

Models the number of successes in $n$ independent trials, each with probability $p$

\[P(X=k) = \binom{n}{k} p^k(1-p)^{n-k}\]

Key Parameters:

$n$ = number of trials
$p$ = probability of success
Mean = $np$
Variance = $np(1-p)$

Examples: A/B test conversions, click-through rates, quality defects

NFL Patriots Coin Toss

The Patriots won 19 out of 25 coin tosses in 2014-15. How likely?

There are 177,100 ways to arrange 19 wins in 25 games
Each specific sequence has probability $0.5^{25}$
Combined probability: 0.5% or odds of 199 to 1 against

Show R code

# "25 choose 19" = number of ways to pick 19 wins from 25 games
choose(25, 19)

[1] 177100

Show R code

# Probability = (ways to get 19 wins) × (probability of any specific sequence)
choose(25, 19) * 0.5^25

[1] 0.005277991

The “Law of Large Numbers” Perspective:

With 32 NFL teams over 20+ years, some team will have a suspicious streak!

Key insight: Probability of Patriots specifically = 0.5%. But probability that some team has a streak ≈ much higher!

Business lesson: When auditing for fraud or anomalies:

Don’t just flag rare events
Consider how many opportunities for rare events exist
Adjust for “multiple comparisons”

Looking at enough data, you’ll always find something “unusual”

Predicting Premier League Goals

How many goals will a team score? Historical EPL data:

Show R code

epl <- read.csv("data/epl.csv")
epl[1:5, c("home_team_name", "away_team_name", "home_score", "guest_score")]

  home_team_name       away_team_name home_score guest_score
1        Arsenal            Liverpool          3           4
2    Bournemouth    Manchester United          1           3
3        Burnley              Swansea          0           1
4        Chelsea             West Ham          2           1
5 Crystal Palace West Bromwich Albion          0           1

Each row = one match with final scores.

The Business Problem:

Sports betting: $200+ billion industry.

Our approach:

Analyze historical data
Model goals as random events
Estimate team strengths
Simulate matches

Who uses this? FiveThirtyEight, ESPN, DraftKings, Betfair, team analytics

EPL Goals: Mean ≈ Variance

A key signature of Poisson data: the mean equals the variance.

Teams score about 1.4 goals per match on average
The variance is also ~1.4 — this is the Poisson fingerprint!
If variance were much larger, we’d need a different model

Show R code

goals <- c(epl$home_score, epl$guest_score)
mean(goals)  # Average goals per team per match

[1] 1.4

Show R code

var(goals)   # Variance ≈ Mean suggests Poisson!

[1] 1.644796

Model Diagnostics: Mean vs Variance

Relationship	Suggests
Variance ≈ Mean	Poisson ✓
Variance > Mean	Overdispersion (Negative Binomial)
Variance < Mean	Underdispersion (rare)

Other Poisson Applications:

Call center arrivals per hour
Website clicks per minute
Insurance claims per year
Manufacturing defects per batch
Emails received per day

Poisson is the “go-to” for count data!

Goals Follow a Poisson Distribution

Show R code

goals <- c(epl$home_score, epl$guest_score)
lambda <- mean(goals)
x <- 0:8
observed <- table(factor(goals, levels = x)) / length(goals)
expected <- dpois(x, lambda = lambda)

barplot(rbind(observed, expected), beside = TRUE, 
        names.arg = x, col = c("steelblue", "coral"),
        xlab = "Goals Scored", ylab = "Proportion",
        legend.text = c("Observed", "Poisson Model"))

Model Validation:

The Poisson model (coral bars) fits the observed data (blue bars) remarkably well!

What this tells us:

Goals are indeed rare, independent events
The Poisson assumption is justified
We can use this model for predictions

Slight discrepancy at 0 goals: Real matches have slightly fewer 0-0 draws than Poisson predicts (teams try harder when level!)

Poisson Distribution

Models count of random events: goals, arrivals, defects, clicks

\[P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\]

$\lambda$ (lambda): expected rate of events
Key property: Mean = Variance = $\lambda$
Events occur independently at a constant average rate

Business Applications:

Customer arrivals per hour
Website clicks per day
Manufacturing defects per batch
Insurance claims per year
Server requests per minute

If events are rare and independent, Poisson is your model!

Improving the Model: Team Strength

A single $\lambda$ for all teams is too simple. Better model:

\[\lambda_{ij} = \text{Attack}_i \times \text{Defense}_j \times \text{HomeAdvantage}\]

Attack: How good is team $i$ at scoring?
Defense: How weak is team $j$ at defending?
Home advantage: ~0.4 extra goals at home

This is how real sports analytics works:

Estimate each team’s offensive/defensive strength from historical data
Adjust for home/away effects
Predict expected goals for each team
Use Poisson to generate win/draw/loss probabilities

Same framework applies to:

NBA point spreads
NFL betting lines
Cricket run predictions
Baseball run expectations

Team-Specific $\lambda$: Arsenal vs Liverpool

To predict a specific match, we estimate each team’s scoring rate:

Arsenal’s attack: How many goals do they typically score at home?
Liverpool’s defense: How many goals do they typically concede away?
Adjustment: Scale by league average to get relative strength

For Arsenal vs Liverpool at home, we estimate Arsenal will score about 1.8 goals on average. Liverpool’s away $\lambda$ would be calculated similarly.

Show R code

# Simple estimate: average goals scored and conceded
arsenal_attack <- mean(epl$home_score[epl$home_team_name == "Arsenal"])
liverpool_defense <- mean(epl$home_score[epl$away_team_name == "Liverpool"])
league_avg <- mean(goals)

# Arsenal's expected goals vs Liverpool (simplified)
lambda_arsenal <- arsenal_attack * (liverpool_defense / league_avg)
lambda_arsenal

[1] 1.851998

Monte Carlo Simulation

Once we have $\lambda$ for each team, we can simulate the match thousands of times.

For Arsenal ($\lambda=1.8$) vs Liverpool ($\lambda=1.5$), running 10,000 simulations gives:

Arsenal wins: ~42% of simulations
Draw: ~24% of simulations
Liverpool wins: ~34% of simulations

This is how betting companies set their odds!

Show R code

set.seed(42)
n_sims <- 10000
# Simulate Arsenal vs Liverpool
arsenal_goals <- rpois(n_sims, lambda = 1.8)  # λ for Arsenal
liverpool_goals <- rpois(n_sims, lambda = 1.5) # λ for Liverpool

# Match outcomes
c(Arsenal_Win = mean(arsenal_goals > liverpool_goals),
  Draw = mean(arsenal_goals == liverpool_goals),
  Liverpool_Win = mean(arsenal_goals < liverpool_goals))

  Arsenal_Win          Draw Liverpool_Win 
       0.4538        0.2254        0.3208

Why Monte Carlo?

Each simulation draws random goals from Poisson distributions

Run 10,000 simulations → get probability of each outcome
Can extend to simulate entire season, league standings
Same approach used by betting companies and analytics firms

This is how FiveThirtyEight and bookmakers build their models!

Monte Carlo Applications:

Finance: Option pricing, portfolio risk (VaR)
Insurance: Claim projections, reserve calculations
Operations: Supply chain uncertainty, demand forecasting
Engineering: Reliability analysis, quality control
AI: Reinforcement learning, MCMC for Bayesian inference

When math is too hard, simulate!

Central Limit Theorem (CLT)

The most important theorem in statistics:

The average of many independent random events tends toward a Normal distribution, regardless of the original distribution.

Why it matters: Stock returns, measurement errors, test scores — all tend to be Normal because they’re sums of many small effects.

Practical Implications:

Sample means are approximately Normal (even if data isn’t)
Confidence intervals work because of CLT
A/B testing relies on CLT for significance tests
Quality control uses CLT for process monitoring

Rule of thumb: Sample size ≥ 30 usually sufficient for CLT to kick in

This is why the Normal distribution is everywhere!

CLT in Action: Michigan Election Polls

Suppose the true vote share in Michigan is 51%. What happens when we poll voters?

Each voter is like a coin flip (vote A or B)
Small samples are noisy; large samples converge to the truth
The distribution of poll results becomes Normal

Show R code

set.seed(42)
true_p <- 0.51
# Poll of 10 voters
hist(replicate(1000, mean(rbinom(10, 1, true_p))), breaks = 20,
     main = "Poll: 10 Voters", xlab = "Vote Share", col = "steelblue", 
     freq = FALSE, xlim = c(0.2, 0.8))
abline(v = true_p, col = "red", lwd = 2, lty = 2)
# Poll of 100 voters
hist(replicate(1000, mean(rbinom(100, 1, true_p))), breaks = 20,
     main = "Poll: 100 Voters", xlab = "Vote Share", col = "steelblue", 
     freq = FALSE, xlim = c(0.2, 0.8))
abline(v = true_p, col = "red", lwd = 2, lty = 2)
# Poll of 1000 voters
hist(replicate(1000, mean(rbinom(1000, 1, true_p))), breaks = 20,
     main = "Poll: 1000 Voters", xlab = "Vote Share", col = "steelblue", 
     freq = FALSE, xlim = c(0.2, 0.8))
abline(v = true_p, col = "red", lwd = 2, lty = 2)

Larger samples → tighter Normal distribution around the true value (red line)

Normal Distribution

The “bell curve” — the most important distribution in statistics

The 68-95-99.7 Rule:

68% of data within 1 standard deviation
95% of data within 2 standard deviations
99.7% of data within 3 standard deviations

Why it’s everywhere: Central Limit Theorem guarantees that averages of many random events become Normal

Applications: Quality control, financial risk, test scores, measurement error

Normal: Heights of Adults

Male heights follow a Normal distribution: mean = 70 inches, sd = 3 inches

68% of men are between 67-73 inches (within 1 sd)
The 95th percentile is about 75 inches — only 5% are taller

Show R code

# What proportion are between 67 and 73 inches (+/- 1 sd)?
pnorm(73, mean = 70, sd = 3) - pnorm(67, mean = 70, sd = 3)

[1] 0.6826895

Show R code

# What height is taller than 95% of men?
qnorm(0.95, mean = 70, sd = 3)

[1] 74.93456

R Functions for Normal Distribution:

Function	Purpose	Example
`pnorm()`	Probability ≤ x	P(height ≤ 73)
`qnorm()`	Find percentile	95th percentile
`dnorm()`	Density at x	Height of curve
`rnorm()`	Random samples	Simulate data

Business Applications:

Setting size ranges for products
Establishing “normal” ranges for KPIs
Identifying outliers (> 2-3 sd)
Quality control limits

The 1987 Stock Market Crash: A 5-Sigma Event

How extreme was the October 1987 crash of -21.76%?

Prior to crash: $\mu = 1.2\%$, $\sigma = 4.3\%$ → Z-score = $\frac{-21.76 - 1.2}{4.3} = -5.34$
Under Normal model: probability = 1 in 20 million (once every 130,000 years!)
Yet 5+ sigma events happened in 1987, 2008, and 2020

Conclusion: The model is wrong — stock returns have “fat tails.” Banks using Normal-based VaR dramatically underestimate risk.

Show R code

pnorm(-5.34)  # Probability of -5.34 sigma event

[1] 4.647329e-08

Fat Tails: Reality vs Normal Model

The Problem with Normal Assumptions:

Stock returns have more extreme events than the Normal distribution predicts.

Event	Normal Probability	Actually Happened
1987 Crash (-22%)	1 in $10^{160}$	Yes
2008 Crisis	“Impossible”	Yes
2020 COVID Crash	“Impossible”	Yes

Implications for Risk Management:

VaR models underestimate tail risk
Need “fat-tailed” distributions (t-distribution, etc.)
Stress testing is essential

Linear Regression

What is Regression?

Finding the relationship between variables

\[y = \beta_0 + \beta_1 x + \epsilon\]

$\beta_0$: intercept (baseline value)
$\beta_1$: slope (change in $y$ per unit change in $x$)
$\epsilon$: unexplained variation

Goal: Minimize sum of squared prediction errors

Business Questions Regression Answers:

How much does price affect sales?
What’s the ROI of advertising spend?
How does experience affect salary?
What drives customer lifetime value?
How does weather affect demand?

Regression quantifies relationships and enables prediction.

Simple Example: House Prices

Using Saratoga County housing data, we fit a model:

Price = f(Living Area)

Intercept: Base price of ~$13,000 (land value)
Slope: Each additional square foot adds ~$113 to the price

A 2,000 sq ft house: $13K + (2000 × $113) = $239,000

Show R code

d <- read.csv("data/SaratogaHouses.csv")
model <- lm(price ~ livingArea, data = d)
coef(model)

(Intercept)  livingArea 
 13439.3940    113.1225

Interpreting Coefficients:

Coefficient	Meaning
Intercept ($13K)	Value of land without house
Slope ($113/sqft)	Price increase per sqft

Making Predictions:

\[\text{Price} = 13,439 + 113 \times \text{SqFt}\]

House Size	Predicted Price
1,500 sqft	$183,000
2,500 sqft	$296,000
3,500 sqft	$409,000

Visualizing the Fit

What the plot shows:

Each blue dot is a house
The red line is our prediction
Vertical distance from dot to line = prediction error

Key observations:

Strong positive relationship
More scatter at higher prices (heteroskedasticity)
Some outliers (expensive small houses, cheap large houses)

The line minimizes the sum of squared vertical distances

Google vs S&P 500 (CAPM)

The Capital Asset Pricing Model (CAPM) asks: Does a stock follow the market or beat it?

\[\text{Google Return} = \alpha + \beta \times \text{Market Return}\]

$\beta$ (beta): How volatile is the stock relative to the market?
$\alpha$ (alpha): Does the stock outperform after adjusting for risk?

Show R code

library(quantmod)
getSymbols(c("GOOG", "SPY"), from = "2017-01-01", to = "2023-12-31") |> invisible()
goog <- as.numeric(dailyReturn(GOOG))
spy <- as.numeric(dailyReturn(SPY))
model <- lm(goog ~ spy)
print(model)


Call:
lm(formula = goog ~ spy)

Coefficients:
(Intercept)          spy  
  0.0003211    1.1705808

Google vs S&P 500: CAPM Results

Our Findings:

Beta ($\beta = 1.01$): Google moves 1:1 with market
Alpha ($\alpha \approx 0$): No significant outperformance ($p = 0.06$)

Beta	Interpretation
$\beta < 1$	Less volatile (utilities, healthcare)
$\beta = 1$	Moves with market (index funds)
$\beta > 1$	More volatile (tech, small caps)

Conclusion: Google tracked the market without consistent alpha in 2017-2023. High beta = higher risk, potentially higher reward.

Orange Juice: Price & Advertising

How does advertising affect price sensitivity? We model sales as a function of price and whether the product was featured in ads.

Key finding: The interaction term (log(price):feat) is negative and significant — advertising changes how customers respond to price!

Show R code

oj <- read.csv("data/oj.csv")
model <- lm(logmove ~ log(price) * feat, data = oj)
tidy(model) |> select(term, estimate, p.value) |> kable(digits = 3)

term	estimate	p.value
(Intercept)	9.659	0
log(price)	-0.958	0
feat	1.714	0
log(price):feat	-0.977	0

The Advertising Paradox

Finding: Advertising increases price sensitivity

Condition	Price Elasticity
No advertising	-0.96
With advertising	-0.96 + (-0.98) = -1.94

Why? Ads coincide with promotions → attract price-sensitive shoppers

Key Lessons:

Correlation ≠ Causation: Ads don’t cause sensitivity; they coincide with promotions
Selection effects: Who responds to ads? Price hunters!
Confounding variables: Promotions happen during ad campaigns
Managerial insight: Don’t blame advertising for price sensitivity — it’s the promotion strategy

Always ask: What’s really driving the relationship?

Logistic Regression

From Regression to Classification

What if the outcome is yes/no?

\[P(y=1 \mid x) = \frac{1}{1 + e^{-\beta^T x}}\]

Why not just use linear regression?

Linear regression can predict values < 0 or > 1
Probabilities must be between 0 and 1
Logistic function “squashes” any input to (0, 1)

NBA Point Spread Example

Can Vegas point spreads predict game outcomes? We fit a logistic regression using historical NBA data.

Show R code

NBA <- read.csv("data/NBAspread.csv")
model <- glm(favwin ~ spread - 1, family = binomial, data = NBA)
tidy(model) |> kable(digits = 3)

term	estimate	std.error	statistic	p.value
spread	0.156	0.014	11.332	0

Interpretation: For each additional point in the spread, log-odds of favorite winning increases by 0.16. The p-value < 0.001 confirms spreads are highly predictive.

Making Predictions

Using our model, we can predict win probability for any point spread:

Spread	P(Favorite Wins)
4 points	65%
8 points	78%
12 points	87%

Show R code

predict(model, newdata = data.frame(spread = c(4, 8)), type = "response")

        1         2 
0.6511238 0.7769474

Same approach used for: credit scoring, churn prediction, marketing response, fraud detection — any binary outcome.

Confusion Matrix

How accurate is our model? The confusion matrix shows predictions vs. actual outcomes.

Show R code

pred <- predict(model, type = "response") > 0.5
table(Actual = NBA$favwin, Predicted = as.integer(pred))

      Predicted
Actual   1
     0 131
     1 422

Our model achieves about 66% accuracy — better than a coin flip!

Reading the Matrix:

	Pred: 0	Pred: 1
Actual: 0	TN (correct!)	FP (oops)
Actual: 1	FN (oops)	TP (correct!)

Sports Betting Reality:

66% accuracy sounds good, but…
Vegas takes ~10% commission (“vig”)
Need ~52.4% accuracy just to break even
Edge of 13.6% is excellent if it holds!

But past performance ≠ future results

Understanding the Confusion Matrix

	Predicted: Win	Predicted: Lose
Actual: Win	True Positive (TP)	False Negative (FN)
Actual: Lose	False Positive (FP)	True Negative (TN)

Key Metrics:

Accuracy = (TP + TN) / Total — overall correctness
Precision = TP / (TP + FP) — “Of predicted wins, how many were right?”
Recall = TP / (TP + FN) — “Of actual wins, how many did we catch?”

Caution: Accuracy can mislead! A spam filter predicting “not spam” for everything has 99% accuracy but catches zero spam. Choose metrics based on business costs.

ROC Curve: The Trade-off

Understanding the ROC Curve:

X-axis: False Positive Rate (false alarms)
Y-axis: True Positive Rate (catches)
Diagonal: Random guessing (AUC = 0.5)
Upper-left corner: Perfect classifier

Area Under Curve (AUC):

AUC	Model Quality
0.5	Random (useless)
0.6-0.7	Poor
0.7-0.8	Fair
0.8-0.9	Good
0.9+	Excellent

Choosing the Right Threshold

The optimal threshold depends on business costs:

Fraud detection: Low threshold (catch more fraud, accept false alarms)
Medical screening: Low threshold (don’t miss disease)
Spam filter: Higher threshold (don’t lose important emails)

There is no universal “correct” threshold

Framework for Threshold Selection:

Quantify costs: What’s the cost of FP vs FN?
Calculate expected cost at each threshold
Choose threshold that minimizes total expected cost

Example — Credit Card Fraud:

False Positive cost: $10 (customer inconvenience)
False Negative cost: $500 (fraud loss)
Optimal threshold: Much lower than 0.5!

Let business economics guide your model decisions

Key Takeaways

Summary

Concept	Key Insight
Distributions	Binomial (binary), Poisson (counts), Normal (continuous)
Poisson	Mean = Variance — the fingerprint of count data
Normal	CLT makes it universal for averages
Linear Regression	Coefficients = effect sizes
Logistic Regression	Outputs probabilities for classification
ROC/AUC	Trade-off between false positives and false negatives
Threshold	Business costs should drive the choice

Statistics is the science of decision-making under uncertainty

Supplemental Reading

Online Articles:

The Surprising Power of Online Experiments - HBR
Machine Learning, Explained - MIT Sloan

Key Insight from HBR: A simple A/B test at Bing generated over $100M annually by testing a “low priority” idea

Books for Further Study:

The Signal and the Noise — Nate Silver
Thinking, Fast and Slow — Daniel Kahneman
Naked Statistics — Charles Wheelan
Data Science for Business — Provost & Fawcett

Online Courses:

Andrew Ng’s Machine Learning (Coursera)
Statistical Learning (Stanford Online)
Fast.ai Practical Deep Learning

Other Formats

Chief AI Officer Program: ML Essentials

Probability Distributions

Why Distributions Matter

Binomial Distribution

NFL Patriots Coin Toss

Predicting Premier League Goals

EPL Goals: Mean ≈ Variance

Goals Follow a Poisson Distribution

Poisson Distribution

Improving the Model: Team Strength

Team-Specific \(\lambda\): Arsenal vs Liverpool

Monte Carlo Simulation

Why Monte Carlo?

Central Limit Theorem (CLT)

CLT in Action: Michigan Election Polls

Normal Distribution

Normal: Heights of Adults

The 1987 Stock Market Crash: A 5-Sigma Event

Fat Tails: Reality vs Normal Model

Linear Regression

What is Regression?

Simple Example: House Prices

Visualizing the Fit

Google vs S&P 500 (CAPM)

Google vs S&P 500: CAPM Results

Orange Juice: Price & Advertising

The Advertising Paradox

Logistic Regression

From Regression to Classification

NBA Point Spread Example

Making Predictions

Confusion Matrix

Understanding the Confusion Matrix

ROC Curve: The Trade-off

Choosing the Right Threshold

Key Takeaways

Summary

Supplemental Reading