Introduction to Probability Theory

Review for Exam # 2

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

1. Expectation
   • What is the formula for EX for a discrete and a continous random variable? Give an example.
   • What is the formula for the expectation for a function g(X) of a random variable.
   • Expectation is a linear operator. What does that mean?
   • Give the definition of a median.
   • For a given density function, how would you give a geometric interpretation of the mean? of the median?

2. Conditional probability
   • What is the definition of conditional probability?
   • Is conditional probability a probability? How would you show this?
   • Explain this definition in the case of equally likely events.
   • What is the law of total probability?
   • Specialize this law to the case of a set and it complement.

3. Independence
   
   • What is the definition of the independence of two events A and B?
   • If A and B are independent, then are their complements independent?
   • Extend the definition of independence to more than two events.

4. Bayes formula
   • Derive Bayes formula.
   • Explain the role of the law of total probability in computations using Bayes formula.

5. Counting methods
   • What is the multiplication principle?
   • How does this principle give us the number of permutations of n objects taken k at a time?

6. Combinations
   • Give an example for which the answer is "7 choose 4".
   • What is the formula for n choose k.
   • Explain in simple terms what this means.
   • What is the identity that gives us Pascal's triangle.
   • How is this related the binomial theorem?
   • Give an example using combinations in a problem involving sampling with replacement.
   • Give an example using combinations in a problem involving sampling without replacement.
   • Show how this formula can be used to derive the multinomial coefficients.