Week6-Discrete Probability Distributions

MATH 11000 Week 6.1

Focus Points

  • Probability Distribution Construction: Learn how to construct a probability distribution for a random variable.

  • Statistical Calculations: Calculate the mean, variance, standard deviation, and expected value for discrete random variables.

Probability Distributions Section 6.1

Random Variables

  • Definition: An experiment or observation is any process by which measurements are obtained.

    • Examples: Counting the number of eggs in a bird's nest, measuring daily rainfall.

  • Notation: The variable representing the result of an experiment is commonly denoted as x.

Types of Random Variables

  • Random Variables: A quantitative variable X is a random variable if its value corresponds to a chance or random outcome.

    • Discrete Random Variable: Takes a finite or countable number of values.

    • Continuous Random Variable: Takes any value within a line interval.

  • Importance: Different mathematical techniques are used for discrete and continuous random variables.

Examples of Random Variables

  • Discrete Examples:

    • X = SAT score for a randomly selected student.

    • X = Number of people in a room at a given time.

    • X = Number on the upper face of a die.

  • Continuous Examples:

    • Air pressure in a tire could take any value in a continuous range.

Discrete Random Variables

  • Count-based examples:

    • Number of students in a statistics class (e.g., 15, 25, 50).

  • Non-examples: 25.5 students is not a valid observation.

Continuous Random Variables

  • Derived from measurements on a continuous scale.

    • Example: Air pressure, which can take values like 20.126 psi.

Identifying Discrete vs Continuous Variables

  • Example tasks: Identify if the following are discrete or continuous:

    • a) Time for a student to register (Continuous)

    • b) Bad checks drawn on a bank (Discrete)

    • c) Gallons needed for a 200-mile drive (Continuous)

    • d) Registered voters who voted (Discrete)

Probability Distribution of a Discrete Random Variable

  • Definition: Assignment of probabilities to each value of the random variable.

  • Features:

    1. Probability assigned to each distinct value.

    2. Sum of all probabilities equals 1.

  • Notation: 0 ≤ p(x) ≤ 1 and Σ p(x) = 1.

Example of Boredom Tolerance Test

  1. Dr. Mendoza tested boredom tolerance among participants with scores from 0 to 6.

  2. To find P(3):

    • P(3) = 6000/20000 = 0.3 (30%).

  3. Use relative frequencies to compute probabilities of other scores ensuring mutual exclusivity and sum equals 1.

Graphing Probability Distributions

  • Relative-Frequency Histogram: Represents the probability of scores visually; bar height represents probability received.

  • Area Interpretation: The area of the bar equals the probability, and sum of areas = 1.

Sum of Probabilities Example for Hiring

  • To calculate probability of hiring a candidate with score 5 or 6:

    • P(5 or 6) = P(5) + P(6) = 0.08 + 0.02 = 0.10 (10%).

Cryptanalysis Example with Letter Frequencies

  1. Analysis of letter frequencies in the Oxford Dictionary.

  2. Calculate probabilities for letters; confirm total adds to 1:

    • Probability for vowels P(A or E or I or O or U) = 0.38 (38%).

Probability Distribution Characteristics

  • Thought of as a relative-frequency distribution based on large n.

  • Mean (µ) and Standard Deviation (σ) defined based on population or sample.

Probability Examples Reminder

  • Examples of basic probabilities:

    • Tossing tails with a coin: 1/2

    • Drawing an ace: 4/52

    • Rolling a specific number on dice: 2/6

    • Totaling a number with a pair of dice: 4/36

Random Variables Example: Two Socks Selection

  1. Selecting from five brown and three green socks. Let X represent the count of brown socks selected:

    • P(X=2) = 20/56, P(X=1) = 30/56, P(X=0) = 6/56.

Mean, Variance, Standard Deviation, and Expectation Section 6.2

Weighted Mean Formula

  • Formula:

    • µ = ∑ xw / ∑ w = ∑ xP(x) / ∑ P(x)

  • This reflects finding expected average values based on probabilities.

Standard Deviation and Risk

  • The standard deviation serves as a measure of risk associated with X.

  • Greater standard deviation indicates more deviation from expected value.

Example of TV Influence on Buying Behavior

  • Result interpretation with results treated as a probability distribution.

Calculation of Mean and Standard Deviation for Buying Influence

  • Calculate:

    • σ = 0.375, expected ads seen is about 2.54.

Carnival Coin-Flipping Game Expected Earnings

  1. Pay $2.00 to play, flip three coins, each showing either 0 or 1.

  2. Determine expected earnings vs. game cost.

  3. Probability of winning calculated and evaluated against costs.

Summary of Expected Value Calculation

  • µ = ∑ xP(x) derived from table of coin results.

  • Estimate indicated player should expect a loss of $0.50 if they play the game.

Extra Examples: Tossing Coins and Heads Definition

  • Example: Probability histogram for tossing two coins, calculating probabilities for x = 0, 1, 2 heads.

Multivariate Distributions: Section 6.3

  • Investigating cases of two or more random variables defined in a joint sample space.

  • Bivariate case involving variables like the sum and product of rolls from dice.

Joint Probability Distribution for Discrete Random Variables

  • Conditions for f(x,y) to be a joint probability distribution made clear with definitions.

Example of Joint Probability Verification

  1. Check defined joint probabilities for given data samples.

Mathematical Expectation: Section 6.4

  • Expectation concerning dice game showing the win and loss probabilities evaluated for monetary expectations.

Expected Value of a Game

  1. Analysis of random variable income against costs indicating this is a losing game based on outcomes.

Multivariate Expected Value Calculation

  • Provides methodology on calculating expected values of combined random variables according to joint distribution.

Example Calculation of Expected Value for Z = X+Y

  • Ensures to evaluate sums in the defining range to find total outcomes expected.

Monty Hall Problem Overview

  • Introduces a probability problem scenario illustrating statistical reasoning in game outcomes.