Math 480, Fall 2016, Assignment 2

The beginner...should not be discouraged if...he finds that he does not have the prerequisites for reading the prerequisites.

- P. Halmos

Carefully define the following terms, then give one example and one non-example of each:

1. Conditional probability distribution (on a finite probability space $S$, conditioned on the event $E$).
2. $P(A|B)$ (the probability of $A$ given $B$).
3. Independent events.
4. Independent random variables.
5. Binomial coefficients.
6. Finite probability space $S$ modelling a Bernoulli trial.
7. Finite probability space $S^n$ modelling a run of $n$ independent Bernoulli trials.
8. Number of successes (as a random variable on $S^n$).
9. Expected value (of a real-valued random variable).

Carefully state the following theorems (you do not need to prove them):

1. Conditional probability formula (for $P(A|B)$, in terms of $P(A\cap B)$ and $P(B)$).
2. Binomial theorem.
3. Arithmetic properties of expectation values.
4. Expected number of successes in $n$ Bernoulli trials.
5. Law of large numbers.

Solve the following problems:

1. Section 1.3, problem 4.
2. Section 1.4, problems 1 (the backslash notation means "complementary event") and 3.
3. Section 1.5, problem 2.
4. Section 1.8, problem 7(a).

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