Overview
This lecture introduces discrete probability distribution functions, their defining properties, and illustrates them using real-life examples.
Discrete Probability Distribution Functions
- A probability distribution function defines a particular probability distribution.
- For discrete distributions, each probability is between 0 and 1, inclusive.
- The sum of all probabilities in the distribution is exactly 1.
Example 1: Baby Crying Data
- Let random variable ( X ) = number of nightly times a newborn wakes its mother.
- Possible values for ( X ): 0, 1, 2, 3, 4, 5.
- Example probabilities: ( P(X=0) = 2/50 ), ( P(X=1) = 11/50 ), ( P(X=2) = 23/50 ), etc.
- This example satisfies both discrete distribution properties: all probabilities are in [0,1], and their sum is 1.
Notation Clarification
- Capital X denotes the random variable.
- Lowercase x represents a specific value the random variable may take.
- The event ( X = x ) means the random variable X takes value x.
Example 2: Nancy's Class Attendance
- Let random variable ( X ) = number of days Nancy attends class in a week.
- Possible values for ( X ): 0, 1, 2, 3.
- Probabilities: ( P(0) = 0.01 ), ( P(1) = 0.04 ), ( P(2) = 0.15 ), ( P(3) = 0.80 ).
- These probabilities all lie in [0,1] and sum to 1, confirming a valid discrete probability distribution.
Key Terms & Definitions
- Probability Distribution Function — A function defining the probabilities of all possible outcomes of a random variable.
- Discrete Probability Distribution — A probability distribution where the random variable has countable possible values.
- Random Variable (X) — A variable representing outcomes of a probabilistic experiment.
- Event (X = x) — The outcome where the random variable X equals a specific value x.
Action Items / Next Steps
- Download and review the relevant chapter from the OpenStax Introductory Statistics textbook.