Lecture Notes 3
1
Uniform Bounds
n
i=1
Recall that, if X1 , . . . , Xn Bernoulli(p) and pn = n1
inequality,
2
P(|pn p| > ) 2e2n .
Xi then, from Hoedings
Sometimes we want to say more than this.
Example 1 Suppose that X1 , . . . , Xn have cdf F . Let
1
Fn
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| >
Lecture Notes 1
Brief Review of Basic Probability
1
Probability Review
I assume you know basic probability. Chapters 1-3 are a review. I will assume you have
read and understood Chapters 1-3. If not, you should be in 36-700.
1.1
Random Variables
A random
Lecture Notes 4
Convergence (Chapter 5)
1
Random Samples
Let X1 , . . . , Xn F . A statistic is any function Tn = g(X1 , . . . , Xn ). Recall that the sample
mean is
n
1
Xi
Xn =
n i=1
and sample variance is
2
Sn
1
=
n1
n
(Xi X n )2 .
i=1
Let = E(Xi ) and