Discrete Probability Distributions: Study Notes for Introductory Statistics (Chapter 5)

Discrete Probability Distributions

Discrete Random Variables

Discrete random variables are fundamental in probability theory and statistics. They represent values that are countable and distinct, such as the number of people in a room or the number of times an event occurs.

• Discrete variables take on individually separate and distinct values.

• A random variable assigns a single numerical value, determined by chance, to each outcome of a procedure.

• Examples include the number of people in a class or the number of times you went to the store last week.

Inequality Review

Understanding inequalities is essential for expressing probability statements and calculating probabilities for discrete random variables.

• Inequalities include: "less than", "less than or equal to", "greater than", "greater than or equal to", and "not equal to".

• Proper notation helps clarify which probabilities are included in calculations.

Translating Statements to Inequalities

Translating real-world statements into mathematical inequalities is a key skill for probability calculations.

• "At most 20" → $x \leq 20$

• "At least twelve" → $x \geq 12$

• "No more than five" → $x \leq 5$

• "More than 100" → $x > 100$

• "Eight items or less" → $x \leq 8$

Probability Distributions

A probability distribution describes the likelihood of each possible value of a random variable. For discrete random variables, the distribution is often presented as a table or graph.

• Requirement 1: The random variable is numerical and associated with corresponding probabilities.

• Requirement 2: The sum of all probabilities is one.

• Requirement 3: Each probability is between zero and one, inclusively.

Example Probability Distribution Table

The table below shows a valid probability distribution for a discrete random variable $x$.

x	1	2	3	4	5
P(x)	0.16	0.22	0.28	0.20	0.14

Identifying Probability Distributions

To determine if a table or graph represents a probability distribution, check the three requirements listed above.

• All probabilities must be between 0 and 1.

• The sum of all probabilities must equal 1.

• The random variable must be numerical.

Examples: Validity of Probability Distributions

x	0	1	2	3	4
P(x)	0.0621	0.2500	0.3749	0.2505	0.0625

This table is a valid probability distribution because all probabilities are between 0 and 1 and their sum is 1.

x	0	1	2	3	4
P(x)	0.4755	-0.5472	0.4717	1.2136	-0.6136

This table is not a valid probability distribution because some probabilities are negative or greater than 1.

Additional Examples

Interview Number, x	1	2	3
P(x)	0.3142	0.3784	0.2985

x	Email	In Person	Text Messages	Direct Messages
P(x)	0.0514	0.5686	0.2658	0.1142

Probability Distribution Graphs

Graphs can also represent probability distributions. The requirements are the same: probabilities must be between 0 and 1, and their sum must be 1.

Using Probability Distributions to Find Probabilities

Probability distributions allow us to answer questions about the likelihood of certain outcomes.

• To find the probability of exactly $k$ successes, use $P(x = k)$.

• To find the probability of at least $k$ successes, sum $P(x)$ for all $x \geq k$.

• To find the probability of fewer than $k$ successes, sum $P(x)$ for all $x < k$.

Example Probability Distribution Table

x	0	1	2	3	4	5
P(x)	0.1302	0.3104	0.1699	0.0997	0.2791	0.0107

Sample Probability Questions

• Exactly three out of five: $P(x = 3) = 0.0997$

• Two or four out of five: $P(x = 2) + P(x = 4) = 0.1699 + 0.2791 = 0.4490$

• At least one out of five: $P(x \geq 1) = P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)$

• Fewer than two out of five: $P(x < 2) = P(x = 0) + P(x = 1)$

• More than three out of five: $P(x > 3) = P(x = 4) + P(x = 5)$

• At most four out of five: $P(x \leq 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4)$

Descriptive Statistics for Discrete Distributions

Descriptive statistics summarize the central tendency and spread of a probability distribution. For discrete distributions, the mean, variance, and standard deviation are calculated as follows:

• Mean (Expected Value): $\mu = \sum [x \cdot P(x)]$

• Variance: $\sigma^2 = \sum [x^2 \cdot P(x)] - \mu^2$

• Standard Deviation: $\sigma = \sqrt{\sum [x^2 \cdot P(x)] - \mu^2}$

Example Calculations

• Given a probability distribution table, calculate the mean, variance, and standard deviation using the formulas above.

• Use a calculator or spreadsheet to sum the products of $x$ and $P(x)$ for the mean, and $x^2 \cdot P(x)$ for variance.

Interpreting Unusual Values

To determine if a value is unusual, check if it lies more than two standard deviations above or below the mean.

• Calculate $\mu + 2\sigma$ and $\mu - 2\sigma$.

• If a value falls outside this range, it is considered unusual.

Summary Table: Requirements for a Valid Probability Distribution

Requirement	Description
Numerical Random Variable	The random variable must be numerical and associated with probabilities.
Sum of Probabilities	The sum of all probabilities must equal 1.
Probability Range	Each probability must be between 0 and 1, inclusively.

Additional info: These notes cover the essentials of discrete probability distributions, including definitions, requirements, examples, and calculations for descriptive statistics. The content is suitable for exam preparation in an introductory statistics course.