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 suppose I tell you that the roll was
an even number. Gi
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 twice.
S = cfw_(i, j ) : i, j cfw_1, . . . , 6
Suppose we
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 problems. Some
problems are easy, some are very hard, bu
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 Gaussian Tail Inequality) Let X N (0, 1). Then
P(|X | >
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 function of X and Y is
F (a, b) = P cfw_X a, Y b, < a, b <
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 ] b
Proof:
E [X ] =
x:p(x)>0
xp(x) a
p(x) = a
x:p(x)>0
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
gene, etc).
2. Distance a ball is thrown.
3. Size of an a
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 in the following way:
lim P cfw_x X x + x | y Y y + y
y
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 we have
E [X ]
P cfw_X a
a
Proof: Consider the random v