Final Exam (2 pages) Probability Theory (MATH 235A, Fall 2007) 1. (12 pts) One tosses a fair coin (H=Heads, T=Tails) until the rst appearance of the pattern HT. How many tosses on average is made? 2. (13 pts) Prove that, for a random variable X with nite
MATH 587
Solutions to Assignment 2
October 23, 2003
(1) f 1 ( (C ) is a -algebra containing f 1 (C ), so contains (f 1 (C ). Let B = cfw_B : f 1 (B ) [f 1 (C )]. Then B is a -algebra containing C , so contains (C ). Hence f 1 ( (C ) (f 1 (C ). (2) (a) Sin
MATH 587
Assignment 2
Due October 26, 2004
(1) Let f and g be extended real-valued Borel measurable functions on (, F ), and dene h( ) = = f ( ) g ( ) if A, if Ac ,
where A is any set in F . Show that h is Borel measurable. (2) Let a Rn , and let B be a B
MATH 587
Assignment 2
Due October 26, 2004
(1) Let f and g be extended real-valued functions on (, F ), and dene h( ) = = f ( ) g ( ) if A, if Ac ,
where A is any set in F . Show that h is Borel measurable. (2) Let a Rn , and let B be a Borel subset of Rn
MATH 587
Assignment 2 Solutions
November 1, 2004
(1) If B B(R), then h1 (B ) = [h1 (B ) A] [h1 (B ) Ac ] = [f 1 (B ) A] [g 1 (B ) Ac ] F , so h is measurable. (2) Let B = cfw_B Rn |x + B B(Rn ). Since x + = and x + Rn = Rn , then , Rn B . If B B , then x
MATH 587
Assignment 3
Due November 18, 2004
(1) Let be a measure, and 1 and 2 be signed measures on (, F ). Prove that (a) if 1 and 2 , then (1 + 2 ) . (b) 1 (c) if 1 (d) if 1 |1 | + , 1 1 . and 2 , then 1 2 . and 1 , then 1 = 0.
Note: if 1 , 2 are signed
MATH 587
Assignment 3 Solutions
November 18, 2004
(1) (a) Let (A) = (B ) = 0, |1 |(Ac ) = |2 |(B c ) = 0. Then (A B ) = 0 and |1 |(C ) = |2 |(C ) = 0, and so 1 (C ) = 2 (C ) = 0 for every C (A B )c . Thus 1 + 2 (C ) = 0 for every C (A B )c , so |1 + 2 |(A
MATH 587
Solutions to Assignment 4
December 2, 2003
1. (a) Consider a subsequence f (Xni , Yni , i 1. Since Xni X in P , there is a further subsequence Xnij , j 1 such that Xnij X a.s. Since Ynij Y in P , there is a further subsequence Ynij , j 1 such tha
MATH 587
Assignment 4
Due November 30, 2004
(1) Let X L2 (, F , P ) and let G be a -subalgebra of F . Show that among all G -measurable r.v.s Y L2 (, F , P ), there is a unique (a.s.) r.v. Y0 which minimizes X Y 2 . (2) (a) Suppose that Y1 , Y2 , . . . ar
MATH 587
Solutions to Assignment 4
November 30, 2004
(1) E (Y X )2 = E [(Y E (X |G ) + (E (X |G ) X )]2 = E [Y E (X |G )]2 + E [(Y E (X |G )(E (X |G ) X )] + E [E (X |G ) X )]2 . But E [(Y E (X |G )(E (X |G ) X )] = E [E (Y X |G )(E (X |G ) X )] = E cfw_E
Quiz Probability Theory (MATH 235A, Fall 2007). October 5, 2007 1. Coin tossing A fair coin is tossed until for the rst time the same result appears twice in succession. Find the probability that the experiment ends before the sixth toss. 2. Permutations
MATH 587
Solutions to Assignment 1
October 12, 2004
(1) Dene (A) = A p( ), A . Obviously (A) 0 for all A. Since (cfw_ ) = p( ) < , then is not identically innite. Finally, let An , n 1 be pw disjoint subsets of , and let A = An . n=1 Then (A) = A p( ) = A
MATH 587
Assignment 1
Due October 5, 2004
(1) Let be a countable set, F = P (). Let p( ), be non-negative numbers. Show that P (A) = A p( ) denes a -nite measure on F . (2) Billingsley (page 41) denes a class C of subsets of to be a -system if 1. C , 2. C
Homework 3 Probability Theory (MATH 235A, Fall 2007) 1. Zero-one law. Let A1 , A2 , . . . be independent events. Prove that the events lim supn An and lim inf n An each have probabilities either zero or one. 2. Bounding random variables. Let X be a random
Homework 5 Probability Theory (MATH 235A, Fall 2007) 1. Random series. Let X1 , X2 , . . . be non-negative random variables. Prove the equality
E
n=1
Xn =
n=1
EXn .
Explain in what sense the equality holds. 2. Sharpness of Chebyshevs inequality. For eve
Homework 6 Probability Theory (MATH 235A, Fall 2007) 1. Discrete independent random variables (a) Show that if X and Y are independent, integer valued random variables, then P(X + Y = n) =
k
P(X = k ) P(Y = n k ).
(b) Let X and Y be independent Poisson ra
Midterm Exam Probability Theory (MATH 235A, Fall 2007) 1. (10 pts) Let X be a non-negative random variable, which takes values 0, 1, 2, . . . Prove that
EX =
n=1
P(X n).
2. (15 pts) From a set of n books, a subset is chosen uniformly at random. All subset
Homework 4
Probability Theory (MATH 235A, Fall 2007)
1. Generating random variables with given distributions. Consider
a funciton F : R R that satises:
(i) F is nondecreasing and 0 F (x) 1 for all x;
(ii) F (x) 0 as x and F (x) 1 as x ;
(iii) F is right c