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Last suggested work

Try these 3 problems. Here are the solutions. In addition here is a challenge problem:

Challenge: Take a finite Markov Chain which is irreducible. Show that all states are recurrent.

Easier problem: If E(|X|ⁿ) is finite then so is E(|X|^m) for 0<m<n and both integers.

From the text:

p106 #'s 1, 2, 3 (deal with renewal processes)
p287 #'s 1, 2 (a bit more convergence)

Suggested Work#4

- read all of lecture 5 on the web and the one given in class
- read al of Chapter 4 (you have done this already), Chapter 5 sections 1,2,3,5. Chapter 7 section 5 (this is the normal). Chapter 8 (this is mainly a review)

Problems from the text:

p143: #'s 1, 2, 3
p146: #'s 1, 3, 4, 6 (assume F is strictly increasing), 7(challenge), 9
p149: #1
p135: #'s 2(read), 4, 5, 11(read)

Challenge: Let N(t) be a Poisson process of rate l on t>=o . Suppose you know N(t₀)=N . Let T₁<...<T_N denote the times of these N points. Evaluate the pdf of T_N-T₁ (this will be conditional on the # of points being N up to and including t₀.
Challenge: Let Z be Bernoulli(p) and U be uniform(0,1). Denote the df of Z by F and, for 0<u<1, define F^-1(u)=min{z: F(z)>=u} . Show F^-1(U)=^dZ . Repeat this for any cts rv Z.

Challenge: X, Y independent Poisson's and W=X+Y. Show X|W=n is binomial.

Suggested Work#3

Please try to complete by October 9.

Suggested Work #1

Partial Solutions

- read 1st lecture

- read Chapter 1, sections 2.1-2.3, 2.9, 3.1, 3.2, comment 5 on p45

- read comments 6,7,8 on p16, 2,3 on p22

- read 3.4, 3.5 "lightly"

Solve the following problems:

- #'s 1, 3, 5 on p 16

- #3 on p21

- p 35 : #'s 3,4,11

- # 6 on p 45

- Show that X and Y having finite 2nd moments implies that X+Y has a finite 2nd moment.

- Show X= 0 implies E(X) =0 . Show E(|X|)=0 implies E(X) =0 (Hint: -|X| <= X <= |X| )

- Challenge : X = 0 wp1 implies E(X) =0 .

- Challenge : # 7 on p37 (This is the Cauchy-Schwartz inequality. You can solve it by noting that E[(X+tY)²] >= 0. This is a quadratic in t. Now use the quadratic formula.)

Suggested Work #2

Partial Solutions

- read sections 2.4, 2.8, 3.3 (lightly), 3.4, 3.5, 4.1-4.3, 4.4, 4.5(lightly) , 4.7, 4.8, 5.2 (this is a review of conditional probability)

- read lecture 2 up to but not including linear prediction.

Problems from the text

p32 Challenge #7
p21 #5
p38 #14
p57 #2
p60 #'s 1 &5 (read) , 4, Challenge #6
p65 #'s 1, 2, 3, Challenge #4 (use the multinomial and take a limit as we did in class for the binomial) , 5, 6, 8 (read)
p74 #1
p78 #'s 3, 4

Also

- Let A1, A2,... be events. Show P(A1 or A2 or ...)<= P(Al) +P(A2) +...
- Let X be a k-dimensional rvec. Show X=0 wp1 iff each component is 0 wp1. Now show this remains true even if X has a countably infinite number of components.

Note: This last problem is easily solved one way by the triangle inequality. On the other hand, if X is 0 wpr then so is X^2 and hence each component squared.

Please have this done by October 2nd.