Homework Set No. 1 Probability Theory (235A), Fall 2013
Due: 10/7/13 at discussion section
1. (The total probability formula) If (, F , P) is a probability space and A, B F
are events such that P(B ) = 0, the conditional probability of A given B is denote
Homework Set No. 6 Probability Theory (235A), Fall 2013
Due: 11/12/13 in class
1. Let X and (Xn ) be random variables dened on a probability space (, F , P ). Show
n=1
P
that if Xn X (convergence in probability) then there is a subsequence (Xnk ) such
k=1
Final exam study guide Probability Theory (235A), Fall 2013
The nal exam will be held on Thursday, Dec. 5 from 1:35 to 3:00 in 1344 Storer
Hall. Please come on time! It will be a closed-book exam. The special distributions handout sheet (similar to the ta
MAT235: Discussion 7
1
Characteristic Function
Let (t) = E (eitX ) where i =
1.1
1 denote the characteristic function.
Some Properties
1. is non-negative denite, which is to say that j,k (tj tk )zj zk 0 for all real t1 , t2 , tn
and complex z1 , z2 , , zn
MAT235: Discussion 8
Modes of convergence
Denition: Let X1 , X2 , be random variables on some probability space (, F, P ).
1. Xn a.s. X almost surely if cfw_ : X ( ) X ( ) as n is an event of probability 1.
r
2. Xn r X in r-th mean where r 1 if E |Xn | <
MAT235: Discussion 6
1
1.1
Generating functions
Probability generating function
Probability generating functions of r.v. X is dened as
GX (s) = E (sX ).
and
P (X = k ) = G(k) (0)
. Theorem: If X and Y are independent, then GX +Y (s) = GX (s)GY (s).
Theore