Week 4 Highlights
Sep 17 - Sep 21
Random variables can be either discrete or continuous.
Random variable X is discrete if the range D is a countable subset of the real numbers. X
can also be dened as discrete if the CDF is a step function
Random variabl
Practice Problems 3
Homework problems to review:
2.2, 3.16 (result), 4.7, 4.16, 4.58, 4.56, 5.24, 5.44, HW 11 #1,2,3.
1. Let Yn be a random variable with distribution Binomial(n, p).
(a) Prove that Yn /n converges in probability to p.
(b) Prove that (Yn /
Practice Problems 1
1. You pick an integer at random between zero and 105 (100,000) inclusive. What is the probability that its digits are dierent? (Do not include leading zeros in the numbers).
2. You have two pairs of red socks, three pairs of mauve soc
Week 14-15 Highlights
Nov 26 - Dec 5
Convergence Concepts
Types of convergence:
1. Convergence in Probability: A sequence of random variables, X1 , X2 , . . . , converges in
probability to a random variable X if, for every > 0,
lim P (|Xn X | ) = 0.
n
p
W
Week 12 Highlights
Nov 12 - Nov 16
Derived Distributions
Students t Distribution
Let X1 , . . . , Xn be iid N (, 2 ). What is the distribution of
X
?
S/ n
We can write this:
X
=
S/ n
(X )/(/ n)
(n 1)S 2 / 2 (n 1)
t(n1)
which is a standard Normal divide
Week 11 Highlights
Nov 5 - Nov 9
Inequalities
Inequalities based on Lemma 4.7.1
Lemma 4.7.1 on which some inequalities are based:
Let a and b be positive numbers and let p and q be positive numbers satisifying
11
+ = 1.
pq
Then
1
p
1
q
ap +
bq ab
with equ
Week 8 Highlights
1
Oct 15-Oct 19
Dierentiating under the integral sign
Leibnitzs Rule give us a method for exchanging the order of integration and dierentiation when
the bounds are nite.
Theorem 2.4.1: Leibnitzs Rule If f (x, ), a(), and b() are dierenti
Week 10 Highlights
Oct 29 -Nov 2
Bivariate Transformations
Let X1 and X2 with joint pdf fX1 ,X2 (x1 , x2 ). Let Y = g (X) where g : R2 R2 . That is
Y1 = g1 (X1 , X2 )
Y2 = g2 (X1 , X2 )
We want to nd the joint pdf of (Y1 , Y2 ).
Assumptions we make:
1. Th
Week 9 Highlights
Oct 22-Oct 26
Multiple Random Vectors
Denition An n-dimensional random vector is a function from a sample space S into Rn , ndimensional Euclidean space.
Denition Let (X, Y ) be a discrete random vector. Then the function f (x, y ) from
Week 6/7 Highlights
Oct 1-Oct 12
Gamma Function
Denition
() =
t1 et dt
0
If > 0 the integral is nite
If is a positive integer the integral can be expressed in closed form
( + 1) = () for > 0
(1) = 1
(n) = (n 1)! for n > 0 an integer
(1/2) =
Exponen
Week 2 Highlights
Sep 3 - Sep 7
Theorem 1.2.14
The Fundamental Theorem of Counting. If a job consists of k separate tasks, the ith of which
can be done in ni ways (i = 1, . . . , k ), then the entire job can be completed in n1 n2 nk ways.
Denition An orde
Week 5 Highlights
Sep 24 - Sep 28
Expected Values
Denition If X is a random variable with pdf f then the expected value of X , EX is given by
(Discrete )
EX =
xf (x)
xX
where X =set of mass/discrete points of X .
(Continuous )
EX =
xf (x)dx
The of a fun
Week 3 Highlights
Sep 10 - Sep 14
Conditional Probability and Independence
Denition If A and B are events in , and P (B ) > 0, then the conditional probability of A given
B , written P (A|B ), is
P (A|B ) =
P (A B )
.
P (B )
Theorem 1.3.5
Bayes Rule. Let
Week 1 Highlights
Aug 27-Aug 31
Denition The sample space, , is the set of possible outcomes of an experiment (the text uses
S to denote the sample space).
Denition An event is any subset of (including itself).
Denition The empty set, written as , is the
Practice Problems 2
1. Suppose that P ( = 1) = 0.3 and P ( = 2) = 0.7 and that X | Poisson().
(a) Evaluate P ( = 1|X = 0).
(b) Find Var(X ) and Cov(, X ).
2. Let X be a positive random variable with pdf
fX (x) = ex
x>0
(a) From this standard distribution,