Ch 5 - Discrete Probability Distributions
Page 1: Introduction
Sanduni Palliyage, PhD
Course: MATH 220, James Madison University
Topic: Chapter 5 Discrete Probability Distributions
Semester: Fall 2024
Page 2: Random Variables
Definition: An outcome of a probability experiment represented by a random variable.
Example: Rolling a die produces values 1-6, each with a probability of 1/6.
Notation: Random variables denoted by uppercase letters (e.g., X).
Summary: Random variables are numerical outcomes of a probability experiment.
Page 3: Types of Random Variables
Discrete Random Variables: Can be listed; possible values include whole numbers.
Continuous Random Variables: Values aren't restricted to a list; can take any value in an interval.
Page 4: Examples
Discrete Example: Rolling a die (possible values 1, 2, 3, 4, 5, 6).
Continuous Example: Height of a student (can be any value in a range, like 161.2 cm to 182.88 cm).
Discrete Example: Number of siblings a person has (possible values include 0, 1, 2, ...).
Continuous Example: Electricity usage in a classroom (can take any value).
Page 5: Focus on Discrete Random Variables
Properties of Discrete Probability Distributions:
Probabilities (P(x)) must be between 0 and 1.
The sum of the probabilities must equal 1.
Formula: Let P(x) denote probability for random variable value x;
0 ≤ P(x) ≤ 1
ΣP(x) = 1
Page 6: Example of Probability Distribution
Consider a fair coin tossed twice; let X be the number of heads:
Possible values for X: 0, 1, 2.
Distribution:
P(X = 0) = 0.25 (TT)
P(X = 1) = 0.50 (HT, TH)
P(X = 2) = 0.25 (HH)
Page 7: Validating Probability Distributions
Given distributions:
Valid: All probabilities between 0 and 1, sum equals 1.
Invalid: Identifies non-valid probabilities (-0.30, 1.15).
Invalid: Identifies a different invalid set; some probabilities do not sum to 1.
Page 8: Probabilities and Events
Use rules of probability to compute probabilities involving random variables:
General Addition Rule: P(A or B) = P(A) + P(B) − P(A and B) if A, B are not mutually exclusive.
If mutually exclusive, P(A or B) = P(A) + P(B).
Rule of Complements: P(A') = 1 - P(A).
Page 9: Example with Blood Pressure Patients
Scenario: 4 patients, let X be number with high blood pressure:
Distribution:
P(0) = 0.23
P(1) = 0.41
P(2) = 0.27
P(3) = 0.08
P(4) = 0.01
Find P(2).
Page 10: Finding Probabilities for High Blood Pressure
Find probability that 3 or 4 patients have high blood pressure:
P(3 or 4) = P(3)+P(4) = 0.08 + 0.01 = 0.09.
Page 11: More Than One Patient with High Blood Pressure
Find probability that more than 1 patient has high blood pressure:
P(x > 1) = P(2)+P(3)+P(4) = 0.27 + 0.08 + 0.01 = 0.36.
Page 12: At Least 3 Patients with High BP
Distribution reiterated, find P(x ≥ 3): P(3) + P(4).
Page 13: Less Than 1 Patient with High BP
Find P(x < 1), which is P(0) = 0.23.
Page 14: Not Exactly 3 Patients with High BP
If event A is exactly 3 patients:
P(A') = 1 - P(3) = 1 - 0.08 = 0.92.
Page 15: Connection Between Distributions and Populations
Statisticians study samples drawn from populations;
Random variables represent observed values from populations.
Page 16: Constructing Probability Distribution
Example: Airport parking with different fees.
Values: $1.50, $2.00, $4.00, $4.50.
Page 17: Distribution Construction Step 1
Frequencies discussed:
Covered long-term: 142
Uncovered long-term: 423
Short-term spaces, etc.
Page 18: Distribution Construction Step 2
Calculate probabilities for each parking option based on frequencies.
Page 19: Mean and Expected Value
Mean (Expected Value) measures the center of distribution:
Formula: μx = Σ[x * P(x)].
Page 20: Example - Defective Pixels Mean Calculation
Distribution shows number of defective pixels, calculate
μ = 0 * 0.2 + 1 * 0.5 + 2 * 0.2 + 3 * 0.1 = 1.2.
Page 21: Probabilities of Defective Pixels and Mean
Reiterate calculation for defective pixels mean, confirmed as 1.2.
Page 22: Interpreting the Mean
Importance of mean in repeated experiments and sampling from populations;
Law of large numbers: larger samples lead to sample mean approaching population mean.
Page 23: Variance and Standard Deviation
Variance measures spread in distribution of X;
Formula: σ^2(X) = Σ[(x - μ_X)^2 * P(x)].
Page 24: Variance/Standard Deviation Example
Calculate variance and standard deviation of defective pixels, starting with the mean.
Page 25: Steps for Variance Calculation
Continue variance calculation, find x - μ_X for each value.
Page 26: Continue with Variance Calculation
Compute squared terms and their contributions to variance.
Page 27: Variance Calculation Summary
Calculate final variance contributions.
Page 28: Conclude Variance Calculation
Find sum of contributions for final results.
Page 29: Alternative Method for Variance
Directly compute the sums of squares for variance calculation.
Page 30: Applications of Mean
Mean can predict gains or losses from actions (expected value).
Page 31: Gambling Context
Origins of probability in gambling; roulette example to illustrate expected losses.
Page 32: Roulette Example Breakdown
Show expected loss with straightforward formulas and calculations.
Page 33: Roulette Winning Probability
Analyze probabilities tied to winning and losing.
Page 34: Business Projections
Profit-loss projections based on expected value principles in business.
Page 35: Profit Example from Mining Venture
Explore expected profits/losses from business ventures with distributions.
Page 36: Insurance Premium Calculations
Insurance companies calculate expected profit/loss based on probabilities.
Page 37: Insurance Premium Situational Context
Review cash flows for probabilities tied to policy outcomes.
Page 38: Probability Distribution of Insurance Policy
Define probability distribution reflecting outcomes and expected values.
Page 39: Expected Values of Insurance
Evaluate final expected gains from numerous instances of policy sales.
Page 40-44: In-Class Problems
Activities to reinforce knowledge of probability distributions, mean calculations, expected values, and variance.