Probabilistic Decision Making (1) - Comprehensive Study Notes

Random Variable

• A random variable is the result of a chance event, that you can measure or count. It is denoted by a variable such as $X$.

• It maps outcomes of a random process to numerical values (discrete or continuous).

Uncertainty and Decision Making

• Example: Pricing decisions under uncertainty.

• Consumer’s Willingness To Pay (WTP):

  • WTP = $25{,}000$ with probability $0.5$

  • WTP = $20{,}000$ with probability $0.5$

• This framing treats price sensitivity and demand as probabilistic, guiding expected profitability under uncertainty.

• Implication: decisions depend on the distribution of WTP rather than a single point estimate.

Probability Distribution (illustrative examples)

• Example: Number of times a U.S. adult has been married (random variable $X$).

  • Possible outcomes: $0, 1, 2, 3+$

  • Corresponding probabilities: $P(X=0)=0.31,<br>n P(X=1)=0.52,<br>P(X=2)=0.13,<br>P(X=3+)=0.04$

• The sum of all probabilities equals 1: $<br>P(X=0)+P(X=1)+P(X=2)+P(X=3+)=1.<br>$

• Fair coin example (data-generating process vs. inference):

  • True data-generating process (for a fair coin): $P( ext{Head})=P( ext{Tail})=0.5$

  • Statistical inference (sampling): toss a coin n times and observe a realized sample; the sample proportion of heads varies across samples.

  • Law of Large Numbers (LLN): as $n o \infty$, the sample proportion converges to $0.5$. Intuition: random fluctuations cancel out with more data; larger samples give better estimates of the truth.

Law of Large Numbers (LLN) – intuition and implications

• With repeated sampling, sample statistics stabilize around the true population values.

• Practical takeaway: confidence in estimates improves with larger sample sizes.

Statistical Inference: Sample Proportions in Bellwether Examples

• Visual intuition (proportions of heads in coin tosses) shows that as sample size grows (e.g., $n=1, n=50, n=100, n=1{,}000{,}000$), the observed proportion concentrates around $0.5$.

• Early samples can be far from the true probability, but deviations shrink with larger $n$.

Expected Value (Mean)

• Coin-toss game (X ∈ {Head, Tail}; Head payoff $30, Tail payoff$10):

  • Probabilities: $P( ext{Head})=0.5, P( ext{Tail})=0.5$

  • Payoffs: Head = $30, ext{ Tail} = 10$

  • Expected reward: $E[X] = 30 imes 0.5 + 10 imes 0.5 = 20$

• General definition: for a discrete random variable with outcomes $x$ and probabilities $P(X=x)$,
  $E[X] = \sum_x x \, P(X=x)$

• Terminology:

  • Random variable: $X$ (realized value: $x$)

  • Expectation/Expected value: $E[X]$

Pricing and Market Prediction

• If a contract pays $1$ with probability $p$ and $0$ with probability $1-p$, then its fair price is: $1 \, imes \, p + 0 \, imes \, (1-p) = p$.

• In markets: the price often reflects the probability of the event, i.e., the fair value aligns with the subjectively assigned probability.

Variance (Spread around the Mean)

• Definition: Variance measures how far a set of numbers are spread out from their average value.

• Example: Income by state (illustrative figures): State B has higher variance than State A; e.g., State A vs State B income distributions show greater dispersion in State B.

• Notation: $ext{Var}(X) = E[(X - E[X])^2] = \sum_x (x - E[X])^2 \, P(X=x)$

• A Coin-toss with Head payoff $40$ and Tail payoff $0$ (probabilities $0.5, 0.5$) has:

  • Expected value: $E[X] = 40(0.5) + 0(0.5) = 20$

  • Variance: $ext{Var}(X) = (40-20)^2(0.5) + (0-20)^2(0.5) = 400$

• Example: Two games with equal expected value but different variance

  • Game I: Head = $25$, Tail = $15$; $P=0.5$ for each

  • $E[X] = 25(0.5) + 15(0.5) = 20$

  • $ext{Var}(X) = (25-20)^2(0.5) + (15-20)^2(0.5) = 25$

  • Game II: Head = $40$, Tail = $0$; $P=0.5$ for each

  • $E[X] = 20$

  • $ext{Var}(X) = (40-20)^2(0.5) + (0-20)^2(0.5) = 400$

• Practice cue: use $ext{Var}(X) = E[(X - E[X])^2] = \sum_x (x - E[X])^2 P(X=x)$

Median and Mean (Illustrative class example)

• Mean income for this class (example): about $2{,}371{,}000$

• This is surprisingly high relative to typical year-incomes in many age groups; typical range reported: between $28{,}000$ and $33{,}000$ per year.

• Median income: $39{,}450$ (illustrative)

• The distribution can be plotted on a log scale to show long tails (Mean vs Median can diverge when distributions are skewed).

Expected Value vs Average; Law of Large Numbers in practice

• Focus in this course: the “average” is used to infer the expected value.

• Examples: Did an advertisement increase sales? Is a pricing ending in 9 effective? Compare sales under conditions A vs B.

• Key principle: the sample average converges to the expected value as sample size grows: $\bar{X}_n <br>ightarrow E[X] ext{ as } n <br>ightarrow \infty$.

Key Probability and Algebraic Properties

• Linearity of expectation:

  • $E[X+Y] = E[X] + E[Y]$

  • $E[X + a] = E[X] + a$, for any constant $a$

  • $E[aX] = a \, E[X]$, for any constant $a$

• Variance under linear shifts and scalings:

  • $ext{Var}[X + a] = ext{Var}[X]$

  • $ext{Var}[aX] = a^2 \, ext{Var}[X]$

Discrete vs Continuous Random Variables

• Discrete random variable:

  • Countable set of possible values (e.g., number of customers in an hour, number of rainy days in a year).

• Continuous random variable:

  • Can take infinitely many values (e.g., weight, height, or a continuous percentage).

Binomial Distribution

• Binomial distribution is a discrete distribution.

• Setup: each trial outcome is either success (1) or failure (0).

  • Random variable $X$ represents the sum of the outcomes after $n$ trials: $X = \sum<em>{i=1}^n x</em>i$ with $x_i \in {0,1}$

  • Probability of success on a single trial: $p$

  • Number of trials: $n$

• Notation: $X \sim \mathrm{Binomial}(n, p)$

• Expectation and variance:

  • $E[X] = n p$

  • $ext{Var}(X) = n p (1 - p)$

Binomial Distribution: Examples

• Coin-toss game (first Binomial example):

  • Number of trials: $n = 100$, probability of heads per trial: $p = 0.5$

  • Payoff: Head = $1$, Tail = $0$ per trial; total payoff is the number of heads, i.e., X = \text{#heads}

  • Expected payoff: $E[X] = n p = 100 \times 0.5 = 50$

  • Interpretation: expected number of heads is 50; if payoff is per head, expected payoff is $50\times 1 = 50$ dollars.

• Other binomial-setup examples include:

  • Taco-order: two taco types; probability of spicy pork ordering: $p = 0.30$; with total orders $n$; expected spicy pork sales: $E[X] = n p$.

  • Coffee-order: two types of coffee; probability of iced latte: $p = 0.40$; with total customers $n$; expected revenue or quantity depends on assigned payoff; variance of counts can be computed using binomial variance formula.

• Greens in Regulation (GIR) example:

  • Greens in Regulation over 72 holes modeled with binomial distributions: field ~ 68%, Scottie ~ 70%

  • Visualized distributions show probabilities across the number of GIRs observed; e.g., 0, 20, 40, 60, 72 GIRs, etc.

Normal Distribution

• Normal distribution is a continuous random variable.

• Parameterization: $X \sim N(\mu, \sigma^2)$ where $\mu$ is the mean and $\sigma^2$ is the variance.

• Shape: bell-shaped, symmetric around the mean, single peak.

• Notation examples: $\Phi<em>\mu,\sigma</em>\varepsilon (X)$ or simpler, just the standard normal when standardized.

• Key properties:

  • Total area under the curve equals 1: $\int_{-\infty}^{\infty} f(x) \, dx = 1$.

  • About the empirical rule: P(\mu - \sigma < X < \mu + \sigma) = 0.6827\, (68.27\%) when X ~ N(\mu, \sigma^2).

• Visual representations show how changing $\mu$ and $\sigma^2$ shifts and stretches the curve.

Practical Questions and Summary

• When modeling uncertainty in marketing research, use probabilistic distributions to represent key quantities (e.g., demand, WTP, sales).

• Compare expected values to observed sample averages; use LLN to justify using sample means for inference as sample size grows.

• Distinguish between discrete vs continuous variables, and choose appropriate distributions (Binomial for counts of successes, Normal for measurement-like data with aggregation).

• Remember the core equations:

  • Expected value: $E[X] = \sum<em>x x \, P(X=x)$ (discrete) or $E[X] = \int</em>{-\infty}^{\infty} x \, f_X(x) \, dx$ (continuous)

  • Variance: $\text{Var}(X) = E[(X - E[X])^2] = \sum_x (x - E[X])^2 P(X=x)$ (discrete) or $\text{Var}(X) = E[X^2] - (E[X])^2$

  • Linearity of expectation: $E[X+Y] = E[X] + E[Y]$; scaling: $E[aX] = a \, E[X]$

  • Variance under linear transformations: $\text{Var}(X + a) = \text{Var}(X)$; $\text{Var}(aX) = a^2 \text{Var}(X)$

Connections to foundations and real-world relevance

• These concepts support probabilistic decision making in marketing: pricing under uncertainty, evaluating promotional effects, and risk assessment.

• The LLN justifies using large-sample averages to estimate population-level effects (e.g., effect of an advertisement on sales).

• Binomial and Normal distributions underpin many marketing metrics: conversion counts, defect rates, and aggregate demand as sample sizes grow.

Quick references to formulas (summary)

• Expected value (discrete): $E[X] = \sum_x x \, P(X=x)$

• Expected value (continuous): $E[X] = \int<em>{-\infty}^{\infty} x \, f</em>X(x) \, dx$

• Variance: $\text{Var}(X) = E[(X - E[X])^2] = \sum_x (x - E[X])^2 P(X=x)$

• Linear properties: $E[X+Y] = E[X] + E[Y],\; E[aX] = a E[X],\; E[X+ a] = E[X] + a$

• Variance under scaling and shifts: $\text{Var}(X + a) = \text{Var}(X),\; \text{Var}(aX) = a^2 \text{Var}(X)$

• Binomial: $X \sim \mathrm{Binomial}(n, p),\; E[X] = n p,\; \text{Var}(X) = n p (1-p)$

• Normal: X \sim N(\mu, \sigma^2),\; P(\mu - \sigma < X < \mu + \sigma) = 0.6827\, (68.27\%)

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