Section 3.2 Conditional Probabilities
EXAMPLES:
1. Consider rolling a die. We know that S = cfw_1, 2, 3, 4, 5, 6. Let E be the event that the
outcome is in cfw_2, 3, 6. We know that P (E ) = 1/2. Now
Section 4.1 Random Variables
Oftentimes, we are not interested in the specic outcome of an experiment. Instead, we are
interested in a function of the outcome.
EXAMPLE: Consider rolling a fair die twi
1
Twenty problems in probability
This section is a selection of famous probability puzzles, job interview questions (most hightech companies ask their applicants math questions) and math competition p
Lecture Notes 2
1
Probability Inequalities
Inequalities are useful for bounding quantities that might otherwise be hard to compute.
They will also be used in the theory of convergence.
Theorem 1 (The
Section 6.1 Joint Distribution Functions
We often care about more than one random variable at a time.
DEFINITION: For any two random variables X and Y the joint cumulative probability distribution fun
Section 7.1 Properties of Expectations: Introduction
Recall that we have
E [X ] =
xp(x)
x
in the discrete case and
E [X ] =
xf (x)dx
in the continuous case.
COROLLARY: If P cfw_a X b = 1, then
a E [X
Section 5.1 Continuous Random Variables: Introduction
Not all random variables are discrete. For example:
1. Waiting times for anything (train, arrival of customer, production of mRNA molecule from
ge
Section 6.5 Conditional Distributions: Continuous Case
Let X and Y be continuous random variables with joint density f (x, y ). We want to dene the
density of X conditioned on Y = y . We motivate it i
Section 8.2 Markov and Chebyshev Inequalities
and the Weak Law of Large Numbers
THEOREM (Markovs Inequality): Suppose that X is a random variable taking only non-negative values.
Then, for any a > 0 w