Discrete Random Variables
Mathematical Statistics
ADA University, School of Business
Information Communication Technologies Agency, Statistics Unit
2025-10-18
Overview
Today’s Journey
- 📚 What is Statistics?
- 🎲 Random Variables
- 📊 Discrete Random Variables
- 📈 Probability Distributions
Learning Objectives
- ✅ Understand the role of statistics
- 🔢 Define random variables
- 🎯 Work with discrete distributions
- 🧮 Apply probability functions
Think-Pair-Share: What is Statistics?
🤔 Think (1 minute): In your own words, what does “statistics” mean to you? How do you use it in daily life?
👥 Pair (2 minutes): Share your definition with a neighbor. Find common themes.
🗣️ Share: Let’s hear some definitions before we explore formal ones!
Definitions of Statistics
Webster’s New Collegiate Dictionary
“A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data.”
Stuart and Ord (1991)
“The branch of the scientific method which deals with the data obtained by counting or measuring the properties of populations.”
Definitions of Statistics (cont.)
Rice (1995)
“Essentially concerned with procedures for analyzing data, especially data that in some vague sense have a random character.”
Key Insight
All definitions emphasize data and uncertainty! 🎲
Definitions of Statistics (cont.)
Freund and Walpole (1987)
Statistics encompasses “the science of basing inferences on observed data and the entire problem of making decisions in the face of uncertainty.”
Mood, Graybill, and Boes (1974)
Statistics is “the technology of the scientific method” and is concerned with:
- The design of experiments and investigations
- Statistical inference
The Core Objective
Central Goal of Statistics
The objective of statistics is to make an inference about a population based on information contained in a sample from that population and to provide an associated measure of goodness for the inference.
Random Variable: Foundation
Definition 1: Random Variable
A random variable is a real-valued function for which the domain is a sample space.
In simple terms: A random variable assigns a numerical value to each outcome in a sample space.
\[Y: S \rightarrow \mathbb{R}\]
Random Variables in Economics & Finance
Economic Applications:
- 📊 Number of unemployed individuals in a region
- 💰 Number of customers entering a bank per hour
- 📉 Number of quarters until recession
- 🏢 Number of firms defaulting on loans in a year
Financial Applications:
- 📈 Number of days stock price increases in a month
- 💵 Number of dividend payments in a year
- 🎲 Number of trading days with losses
- 🏦 Number of loan applications approved
Example 1: Investment Returns
Problem Setup
An investor evaluates two stocks and decides to invest in one. Based on market analysis: - Probability both perform well: 0.25 - Probability exactly one performs well: 0.50 - Probability neither performs well: 0.25
Let \(Y\) equal the number of stocks that perform well.
Tasks:
- Identify the sample space
- Assign values of \(Y\) to each outcome
- Find the probability distribution
Example 1: Solution
Step 1: Sample Space
\[S = \{BB, BG, GB, GG\}\] where B = bad performance, G = good performance
Step 2: Probability Distribution
| \(Y\) | Sample Points | \(P(Y = y)\) |
|---|---|---|
| 0 | {BB} | 0.25 |
| 1 | {BG, GB} | 0.50 |
| 2 | {GG} | 0.25 |
Example 1 (Alternative): Coin Toss
Problem Setup
Consider an experiment as tossing two coins and observing the results. Let \(Y\) equal the number of heads obtained.
Tasks:
- Identify the sample points in \(S\)
- Assign a value of \(Y\) to each sample point
- Identify the sample points associated with each value of \(Y\)
Example 1 (Alternative): Solution
Step 1: Sample Space
\(S = \{HH, HT, TH, TT\}\)
Step 2: Assign Values
| Sample Point | \(Y\) (Number of Heads) |
|---|---|
| HH | 2 |
| HT | 1 |
| TH | 1 |
| TT | 0 |
Example 1: Solution (cont.)
Step 3: Values of Random Variable
| \(Y\) | Sample Points | \(P(Y = y)\) |
|---|---|---|
| 0 | {TT} | 1/4 |
| 1 | {HT, TH} | 2/4 = 1/2 |
| 2 | {HH} | 1/4 |
Visualization:
Key Insight
Remark 1
\(P(Y = y)\) is the sum of the probabilities of the sample points that are assigned the value \(y\).
Example:
\[P(Y = 1) = P(\{HT\}) + P(\{TH\}) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}\]
🎯 Understanding Check: Three Coins
If we toss three coins, how many possible values can the random variable “number of heads” take?
- 2 values
- 3 values
- 4 values
- 8 values
Random Sampling
Definition 2: Random Sample
Let \(N\) and \(n\) represent the numbers of elements in the population and sample, respectively.
If the sampling is conducted in such a way that each of the \(\binom{N}{n}\) samples has an equal probability of being selected, the sampling is said to be random, and the result is said to be a random sample.
Random Sampling: Importance
Remark 2
The method of sampling, known as the design of an experiment, affects both:
- The quantity of information in a sample
- The probability of observing a specific sample result
Hence, every sampling procedure must be clearly described if we wish to make valid inferences from sample to population.
Think-Pair-Share: Sampling Methods
🤔 Think (1 minute): You need to survey 100 students from a university of 10,000 students. What methods could you use? Which would give a random sample?
👥 Pair (2 minutes): Discuss potential biases in different sampling methods.
🗣️ Share: What are the pros and cons of each method?
Discrete Random Variable
Definition 3: Discrete Random Variable
A random variable \(Y\) is said to be discrete if it can assume only a finite or countably infinite number of distinct values.
Examples in Finance/Economics:
- Number of defaults in a loan portfolio (finite)
- Number of trades executed per day (finite but large)
- Number of customers served until first complaint (countably infinite)
- Number of economic shocks in a year (finite)
- Number of dividend payments in a year (finite)
Notation
Standard Notation
- \(Y\) : random variable (capital letter)
- \(y\) : particular value (lowercase letter)
- \((Y = y)\) : the set of all points in \(S\) assigned the value \(y\) by the random variable \(Y\)
- \(P(Y = y)\) : the probability that \(Y\) takes on the value \(y\)
Probability Function
Definition 4: Probability Function
The probability that \(Y\) takes on the value \(y\), \(P(Y = y)\), is defined as the sum of the probabilities of all sample points in \(S\) that are assigned the value \(y\).
We will sometimes denote \(P(Y = y)\) by \(p(y)\), which is called probability function for \(Y\).
\[p(y) = P(Y = y) = \sum_{\omega \in S: Y(\omega) = y} P(\{\omega\})\]
Probability Distribution
Definition 5: Probability Distribution
The probability distribution for a discrete variable \(Y\) can be represented by a formula, a table, or a graph that provides \(p(y) = P(Y = y)\) for all \(y\).
Three Representations:
- Formula: \(p(y) = \frac{1}{6}\) for \(y = 1, 2, 3, 4, 5, 6\) (fair die)
- Table: Listing all \(y\) values with their probabilities
- Graph: Bar chart showing probability distribution
Example 2: Worker Selection
Problem Setup
A supervisor in a manufacturing plant has three men and three women working for him. He wants to choose two workers for a special job.
Not wishing to show any biases in his selection, he decides to select the two workers at random.
Let \(Y\) denote the number of women in his selection.
Task: Find the probability distribution for \(Y\).
Example 2: Solution - Setup
Step 1: Identify Possible Values
\(Y\) can take values: 0, 1, or 2 (number of women selected)
Step 2: Total Number of Ways to Select 2 Workers
Total workers = 6 (3 men + 3 women)
\[\text{Total ways} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15\]
Example 2: Solution - Calculations
Case 1: \(Y = 0\) (no women, 2 men)
\[P(Y = 0) = \frac{\binom{3}{2} \times \binom{3}{0}}{\binom{6}{2}} = \frac{3 \times 1}{15} = \frac{3}{15} = \frac{1}{5}\]
Case 2: \(Y = 1\) (1 woman, 1 man)
\[P(Y = 1) = \frac{\binom{3}{1} \times \binom{3}{1}}{\binom{6}{2}} = \frac{3 \times 3}{15} = \frac{9}{15} = \frac{3}{5}\]
Case 3: \(Y = 2\) (2 women, no men)
\[P(Y = 2) = \frac{\binom{3}{2} \times \binom{3}{0}}{\binom{6}{2}} = \frac{3 \times 1}{15} = \frac{3}{15} = \frac{1}{5}\]
Example 2: Probability Distribution
Table Representation:
| \(y\) | \(p(y) = P(Y = y)\) |
|---|---|
| 0 | 1/5 = 0.20 |
| 1 | 3/5 = 0.60 |
| 2 | 1/5 = 0.20 |
| Sum | 1.00 |
Verification: \(\frac{1}{5} + \frac{3}{5} + \frac{1}{5} = \frac{5}{5} = 1\) ✅
Example 2: Visual Distribution
Properties of Probability Distributions
Theorem 1: Fundamental Properties
For any discrete probability distribution, the following must be true:
\(0 \leq p(y) \leq 1\) for all \(y\)
\(\sum_{y} p(y) = 1\)
where the summation is over all values of \(y\) with nonzero probability.
Why are these important?
- Property 1: Probabilities must be between 0 and 1
- Property 2: The total probability across all outcomes must equal 1 (certainty)
Group Activity: Verify Properties
Activity (3 minutes)
In small groups, verify that Example 2 (worker selection) satisfies both properties of Theorem 1.
Check:
- Is \(0 \leq p(y) \leq 1\) for all \(y\)?
- Does \(\sum_{y} p(y) = 1\)?
Solution:
- \(p(0) = 0.20\), \(p(1) = 0.60\), \(p(2) = 0.20\) — all between 0 and 1 ✅
- \(0.20 + 0.60 + 0.20 = 1.00\) ✅
Practice Problem: Portfolio Risk
Your Turn!
An investment portfolio contains 4 high-risk and 2 low-risk assets. A portfolio manager randomly selects 3 assets for quarterly review.
Let \(Y\) = number of high-risk assets selected.
Tasks:
- What are the possible values of \(Y\)?
- Find \(P(Y = y)\) for each value
- Verify Theorem 1
Practice Problem: Solution Setup
Step 1: Possible Values
\(Y\) can be: 1, 2, or 3 (need at least 1 high-risk since only 2 low-risk assets)
Step 2: Total Ways to Select 3 from 6
\[\binom{6}{3} = \frac{6!}{3!3!} = 20\]
Practice Problem: Solution
Case 1: \(Y = 1\) (1 high-risk, 2 low-risk)
\[P(Y = 1) = \frac{\binom{4}{1} \times \binom{2}{2}}{\binom{6}{3}} = \frac{4 \times 1}{20} = \frac{4}{20} = 0.20\]
Case 2: \(Y = 2\) (2 high-risk, 1 low-risk)
\[P(Y = 2) = \frac{\binom{4}{2} \times \binom{2}{1}}{\binom{6}{3}} = \frac{6 \times 2}{20} = \frac{12}{20} = 0.60\]
Case 3: \(Y = 3\) (3 high-risk, 0 low-risk)
\[P(Y = 3) = \frac{\binom{4}{3} \times \binom{2}{0}}{\binom{6}{3}} = \frac{4 \times 1}{20} = \frac{4}{20} = 0.20\]
Practice Problem: Distribution
| \(y\) | \(p(y)\) |
|---|---|
| 1 | 0.20 |
| 2 | 0.60 |
| 3 | 0.20 |
| Sum | 1.00 |
Verification:
- All probabilities \(\in [0,1]\) ✅
- Sum = 0.20 + 0.60 + 0.20 = 1.00 ✅
📊 Interactive Quiz: Valid Probability Distribution
Which of the following could be a valid probability distribution for a discrete random variable \(Y\)?
- \(P(Y=1) = 0.3, P(Y=2) = 0.5, P(Y=3) = 0.3\)
- \(P(Y=1) = 0.2, P(Y=2) = 0.5, P(Y=3) = 0.3\)
- \(P(Y=1) = -0.1, P(Y=2) = 0.6, P(Y=3) = 0.5\)
- \(P(Y=1) = 0.4, P(Y=2) = 0.4, P(Y=3) = 0.4\)
Applications in Economics & Finance
Why Discrete Random Variables Matter
Economics:
- Labor market analysis (employment status)
- Consumer choice models (number of purchases)
- Market structure (number of firms)
- Policy evaluation (number of beneficiaries)
Finance:
- Risk management (number of defaults)
- Portfolio optimization (asset allocation)
- Derivative pricing (binomial trees)
- Credit scoring (risk categories)
Key Takeaways
Summary
Statistics is about making inferences from samples to populations
Random variables map sample space outcomes to numerical values
Discrete random variables take finite or countably infinite values
Probability distributions must satisfy:
- \(0 \leq p(y) \leq 1\) for all \(y\)
- \(\sum_{y} p(y) = 1\)
Applications span economics, finance, and business decision-making
Homework Problems
Problem 1: Exchange Rate Analysis
A currency can move Up, Down, or Unchanged daily. Historical data shows: P(Up)=0.4, P(Down)=0.3, P(Unchanged)=0.3.
Let \(Y\) = number of “Up” days in 2 trading days.
- Find the probability distribution of \(Y\)
- What is the most likely outcome?
- Verify properties of probability distributions
Homework Problems (cont.)
Problem 2: Loan Portfolio
A bank’s portfolio has 6 loans: 4 are performing well, 2 are underperforming. A regulator randomly audits 3 loans.
Let \(Y\) = number of underperforming loans in the audit.
- Find \(P(Y = y)\) for all values
- What is \(P(Y \geq 1)\)?
- Graph the probability distribution
Homework Problems (cont.)
Problem 3: Financial Transaction Model
Consider the probability function:
\[p(y) = \begin{cases} cy^2 & \text{for } y = 1, 2, 3 \\ 0 & \text{otherwise} \end{cases}\]
- Find \(c\) that makes this valid
- Calculate \(P(Y \leq 2)\)
- Calculate \(P(Y > 1)\)
- Interpret \(Y\) in a financial context (e.g., number of high-value transactions)
Homework Solution Hints
Problem 1 Hint:
List all possible outcomes for 2 days: (U,U), (U,D), (U,N), etc. Calculate probability for each outcome using independence.
Problem 2 Hint:
Use hypergeometric probability: \(P(Y=y) = \frac{\binom{2}{y}\binom{4}{3-y}}{\binom{6}{3}}\)
Problem 3 Hint:
Use property \(\sum_{y=1}^{3} cy^2 = 1\) to solve for \(c\). Then \(c(1^2 + 2^2 + 3^2) = c(14) = 1\).
Next Class Preview
Coming Up Next
- Expected Value and Variance of discrete random variables
- Common discrete distributions: Binomial, Poisson, Geometric
- Applications to real-world problems
Preparation: Review the concept of weighted averages and summation notation.
Questions?
Thank you!
Office Hours: By appointment via email
Contact: sorujov@ada.edu.az
Mathematical Statistics - Discrete Random Variables