Chapter 4: Discrete Probability Distributions

Section 4.1: Probability Distributions

This section introduces the concept of probability distributions for discrete random variables, including how to construct, verify, and analyze such distributions. It also covers the calculation of mean, variance, standard deviation, and expected value for discrete probability distributions.

Random Variables

• Random Variable: A variable that represents a numerical value associated with each outcome of a probability experiment. Denoted by x.

• Discrete Random Variable: Has a finite or countable number of possible outcomes that can be listed. Example: Number of sales calls a salesperson makes in one day.

• Continuous Random Variable: Has an uncountable number of possible outcomes, represented by an interval on the number line. Example: Hours spent on sales calls in one day.

Identifying Discrete and Continuous Variables

• Discrete: If the variable can be counted (e.g., number of Fortune 500 companies that lost money).

• Continuous: If the variable can take any value within an interval (e.g., volume of gasoline in a tank).

Discrete Probability Distributions

• Definition: A discrete probability distribution lists each possible value the random variable can assume, together with its probability.

• Conditions:

  • Each probability must be between 0 and 1, inclusive.

  • The sum of all probabilities must equal 1.

Constructing a Discrete Probability Distribution

1. Make a frequency distribution for the possible outcomes.

2. Find the sum of the frequencies.

3. Calculate the probability of each outcome by dividing its frequency by the total frequency.

4. Verify that each probability is between 0 and 1, and the sum of all probabilities is 1.

Example: Constructing and Graphing a Discrete Probability Distribution

• An industrial psychologist administered a personality inventory test for passive-aggressive traits to 150 employees. Scores ranged from 1 (extremely passive) to 5 (extremely aggressive).

• To construct the probability distribution, divide the frequency of each score by the total number of individuals.

• Graph the distribution using a histogram, where the area of each bar represents the probability of each outcome.

• The distribution is approximately symmetric.

Verifying a Probability Distribution

• Check that each probability is between 0 and 1.

• Check that the sum of all probabilities equals 1.

• If either condition fails, it is not a valid probability distribution.

• Example: If the sum of probabilities is 1.07, it is not a probability distribution. If any probability is negative or greater than 1, it is not a probability distribution.

Mean of a Discrete Probability Distribution

• Definition: The mean (expected value) is calculated by multiplying each value of x by its corresponding probability and summing the products.

• Formula:

• Example: For the personality inventory test, the mean score is slightly less than 3, indicating a tendency toward passive traits.

Variance and Standard Deviation

• Variance: Measures the spread of the distribution.

• Standard Deviation: The square root of the variance.

• Formulas:

  • Variance:

  • Standard Deviation:

• Example: Most data values differ from the mean by no more than 1.3.

Expected Value

• Definition: The expected value of a discrete random variable is equal to its mean.

• Formula:

• Example: In a raffle with 1500 tickets sold at $2 each for four prizes, the expected value for one ticket is calculated by subtracting the ticket price from each prize and multiplying by the probability of winning each prize. The expected value is negative, indicating an average loss per ticket.

Summary Table: Properties of Discrete Probability Distributions

Additional info: The image included is the cover of the textbook "Elementary Statistics" by Ron Larson, which is directly relevant as it visually identifies the source and context of the study material.