Lectures and material in the absence of Professor Liu
I will use the handouts that I have given to Dr Liu plus some additional material on the blackboard
I will give Dr Liu's lectures in this course until October 31, 2016.
Office hours : These will be irregular for this time period.
Oct 18 2016 : I will be at my office 10-11 AM.
Oct 20 2016 : I will be at my office 10-11 AM, but I am the local host for a colloquium (academic visitor) who will arrive between 10:30 to noon), and will end the office hour at that time.
Friday Oct 21 : afternoon between 2 and 3 PM
Week of Oct 24-28 to be announced. There is a 2/5-th reading week (2 out of the 5 week days) on Oct 27 and 28.
October 17, 2016
Change of variables formula for double integrals with invertible differentiable transforms
AETaylor-AdvancedAclculusIntegralTransformTheorem.pdf from from A E Taylor, Advanced Calculus, Blaisdell Publishing Company, Toronto, 1955. There are newer editions, Taylor and Mann 3rd Edition, or many other Advanced Calculus texts for multivariable change of variables of integration.
Study of Transformations (Section 3.6 of Rice)
convolutions and ratios
general mappings - here one needs to be careful of properties of the mapping ie the function and then apply ideas from the Axioms of Probability. This forces the type of mathematical tools that apply
One can sometimes use a completion of a mapping when one has a many to one mapping
October 19, 2016 :
Preliminary exam timetable.
Room to be announce in final exam schedule.
continue with uniform ratios example.
Brief discussion of discrete cases. S3-6DiscreteExample.pdf
Some suggested problems : from Section 3.8
36 (can you use one of Propositions 2.3C or D?), 42a, 51, 52, 63, 64
Section 3.7 : Order statistics is a mapping that is many to one. Sometimes our tools as already studied apply. Sometimes these tools need extra work, in higher dimensions. Returning to an idea related to the Fundamental Theorem of Calculus ( a higher dimensional analogue) is also useful.
orderstats.pdf (two typos corrected for example 1, 21 Oct 2016).
Suggested problem : X_1, ... , X_3 iid exponential mean 1. Find the pdf of X_(2). Notice here your integrand will be a sum of various exponentials.
Section 3.8 : 72 , 74
After discussing order statistics we move on to Chapter 4, expectation and conditional expectation and some related topics (moment generating functions).
October 24, 2016 : Ch4_Expectation.pdf (includes Chebyshev's inequality)
October 26, 2016 : continue with expectations and time start Ch4-4-cond-expec.pdf (material from Section 4.4)
Suggested problems Section 4.7 : 8, 13, 18 (do the expectation by linearity and also be finding the distribution of U_(n) - U_(1) )
31, 34, 43, 44, 53, 60, 62, 67, 70, 77
Suggested problems for additional material covered in Chapter 4.
Section 4.7 : 88, 89, 93, 98, 100, 101 , 103
MGF.pdf Moment generating functions
Moment approximation Momapprox.pdf (section 4.6) which also includes some material on construction of QQ plots which is not covered in the text.
MONDAY OCT 31, 2016 : Dr Stanford has sent out an announcement about grading the first term test. All registered students should have received this by now. He will post the grades by the end of today (before midnight).