Introduction to Probability Theory

Review for Exam # 1

Close all books and notebooks and ask yourselves the following questions.

1. Experiments, sample space, outcomes, events
   • What is an experiment? Give an example.
   • Given an experiment, how do you find its sample space?
   • What is an outcome?
   • Can a sample space contain an infinite number of outcomes? Why or why not?
   • Is there a difference between an outcome and an event? Give examples.

2. Set theory
   • In a word problem, how would you recognize that you have to find
     • the union
     • the intersection
     of two events?
   • In a word problem, how would you recognize that you have to find the complement of one event?
   • What are the properties of the union, the intersection and the complement?
   • What does it mean that two events are mutually exclusive?

3. Definition of probability
   
   • What are the 3 axioms of probability?
   • What is the probability of
     • S ?
     • The null set?
   • What is the formula for the probability of the union of two events?
   • Can you explain what this formula tells you in terms of Venn diagrams?
   • What is the formula for the probability of the union of three events?
   • What does it mean that two events are equally likely?

4. Random variables
   • What is a random variable? Give an example.
   • What is the difference between a discrete and a continuous random variable? Give examples.
   • What is a Bernoulli random variable?
   • What is a binomial random variable?
   • What is the probability function of a discrete random variable? What kind of information does it give you? What are its properties? Give an example.
   • What is the probability density function of a random variable? What kind of information does it give you? What are its properties? Give an example.
   • What is the difference between a probability density function and a probability function ?
   • What is the cumulative distribution function of a random variable? What kind of information does it give you? What are its properties?
   • How would you be able to tell whether a given function could or could not be
     • a probability function?
     • a probability density function?
     • a cumulative distribution function?
   • How would you find the cumulative distribution function of a random variable if you were given
     • The graph of its probability function or probability density function?
     • A formula for its probability function or probability density function?
   • How is the mean of a random variable defined? Give formulas.
   • How would you estimate the mean of a random variable from the graph of its probability density function or probability function?
   • Can a random variable have more than one mean?
   • What is a median of the distribution of a random variable? How is it defined? How can there be more than one median? Give examples.
   • How do you calculate the expectation of a function of a random variable?
   • Sketch the probability function of a finite random variable. Label the axes and use this information to find the expectation and a median of this random variable.