Homework Set No. 9 Probability Theory (235A), Fall 2011
Due: 11/29/09
1. (a) If X is a r.v., show that Re(X ) (the real part of X ) and |X |2 = X X are
also characteristic functions (i.e., construct r.v.s Y and Z such that Y (t) = Re(X (t),
Z (t) = |X (t)
Homework Assignment #4
Math 235B
UC Davis, Winter 2012
Homework due: Monday 3/19/12 at noon. Submit by email or by bringing the HW
to my oce, MSB 2218 (slide it under the door if Im not there).
Problems
1. Prove that entropy is a concave function of proba
Homework Set No. 8 Probability Theory (235A), Fall 2011
Due: 11/22/11
1. (a) Let X1 , X2 , . . . be a sequence of independent r.v.s that are uniformly distributed
on cfw_1, . . . , n. Dene
Tn = mincfw_k : Xk = Xm for some m < k .
If the Xj s represent the
Homework Assignment #1
Homework due:
Math 235B
UC Davis, Winter 2012
Tuesday 2/7/12
Problems
1. Let (, F , P) be a probability space, X be a random variable on the space, and
G1 G2 F be two sub- -algebras of F . Prove that
E (E (X | G2 ) | G1 ) = E (X | G
Homework Assignment #3
Homework due:
Math 235B
UC Davis, Winter 2012
Thursday 3/8/12 (note extended deadline)
Problems
1. (a) Let T : (0, 1) (0, 1) be a piecewise-smooth and piecewise strictly monotonic interval
map. Let be a probability measure on (0, 1)
Homework Assignment #2
Homework due:
Math 235B
UC Davis, Winter 2012
Tuesday 2/21/12
Problems
1. Recall that if X1 , X2 , . . . is a sequence of random variables, an event A is called a tail event
for the sequence (Xn )n if for any n 1, A (Xn+1 , Xn+2 , .