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ISE 3414
Topic #22: Expectation and Variance for a Compound Random Variable
Definition: If
are a sequence of IID random variables, and N is a non-negative
integer-valued random variable that is independent of the s, then
is called a
compound random variab
ISE 3414
Topic #21: Computing Probabilities by Conditioning
We can use also conditioning to compute
, where X and Y are both random
variables. Again, the concepts are rather intuitive, so I think we can develop a solution
approach without the formulas.
Ex
ISE 3414
Topic #15: Central Limit Theorm
Variance for the Sum of Random Variables
In Topic #13, we showed that Var[X + Y] = Var(X) + Va'r(Y) + 2C0v(X, Y).
We can also extend this concept to summations of more than two random variables.
Var[ {21Xi] = Var(X
ISE 3414
Topic #13: Covariance
We have already discussed how to take the expected value of the sum of two random
variables. We would now like to address the variance of a sum of random variables.
However, we must first define the concept of covariance.
Co
ISE 3414
Topic #10: Joint Continuous Random Variables
We now consider joint distributions for continuous random variables. The concepts
parallel those we developed for joint discrete distributions, where we replace addition by
integration.
Join
ISE 3414
Topic #29: Continuous-Time Markov Chains
Chapter 6: Continuous-Time Markov Chains
We now address the class of Markov chains where the time index takes on continuous
values (as opposed to Chapter 4 where state changes could only occur at discrete
ISE 3414
Topic #18: Conditional Distributions and Expectations
Chapter 3: Conditional Probability and Expectation
We now will dive more deeply into some concepts about conditional probability. We will
start by reviewing the basic concept of conditional di
ISE 3414
Topic #4: Basic Probability
Now that we know how to count outcomes, we can start talking about calculating
probabilities.
Probability: The likelihood of the occurrence of an event.
Note: Every sample point has a weight (or probability) associated
ISE 3414
Topic #12: Variance
It is not enough to just know average behavior. The range of outcomes matters as well.
Consider the following three games as an example.
Game X: Win $0 with probability 1.
Game Y: Win $1 with probability and lose $1 with proba
ISE 3414
End-of-Course Wrap-Up
Hopefully, by this point in the summer, you feel like youve learned a lot about probability!
If you need some convincing, heres a summary of the various things weve covered.
Probability Basics
o Counting sample points
o Prob
ISE 3414
Topic #11: Expected Values
In order to summarize how different types of distributions behave, we now consider the
concept of an expected value for a random variable. Expected values represent the
average behavior of the random variable.
Consider
ISE 3414
Topic #9: Joint Discrete Random Variables
Previously, weve dealt with the probability that one event occurred, e.g. P(a < X < b).
Now we consider the probability that two or more events occur simultaneously, e.g.
P(a<X<b, c<Y<d).
Ioint Probabilit
ISE 3414
Midterm Exam #1
Tuesday, June 11th, 2013
Instructions:
1. This exam contains a total of 6 problems on 7 pages, for a total of 100 points. There
is also a 5 point bonus question. Please check to make sure that you have all the
questions.
2. You ar
ISE 3414
Midterm Exam #1
Tuesday, June 5th, 2012
Instructions:
1. This exam contains a total of 5 problems on 6 pages, for a total of 100 points. There
is also a 5 point bonus question. Please check to make sure that you have all the
questions.
2. You are
ISE 3414
Midterm Exam #2
Tuesday, June 21st, 2011
Instructions:
1. This exam contains a total of 6 problems for a total of 100 points (plus one bonus
problem) on 8 pages. Please check to make sure that you have all the questions.
2. You are permitted to u
\
1, {T pits.) Khalif and Kam'm (bath in an: POR class) hike Emu: Dragons Tooth to McMaei
Knob (two very popular tails) every Saturday morning dining the Fall Selez. Along
this route, they tak: a break when they reach the middle point (point B). If the we
\
1, {T pits.) Khalif and Kam'm (bath in an: POR class) hike Emu: Dragons Tooth to McMaei
Knob (two very popular tails) every Saturday morning dining the Fall Selez. Along
this route, they tak: a break when they reach the middle point (point B). If the we
ISE 3414
Midterm Exam #2
Tuesday, June 25th, 2013
Instructions:
1. This exam contains 6 problems, plus a bonus question, on a total of 8 pages. Please
check to make sure that you have all the questions.
2. A cumulative normal table is attached to the back
ISE 3414
Midterm Exam #2
Tuesday, June 19th, 2012
Instructions:
1. This exam contains 6 problems, plus a bonus question, on a total of 8 pages. Please
check to make sure that you have all the questions.
2. A cumulative normal table is attached to the back
ISE 3414
Midterm Exam #2
Tuesday, June 19th, 2012
Instructions:
1. This exam contains 6 problems, plus a bonus question, on a total of 8 pages. Please
check to make sure that you have all the questions.
2. A cumulative normal table is attached to the back
ISE 3414
Midterm Exam #2
Tuesday, June 25th, 2013
Instructions:
1. This exam contains 6 problems, plus a bonus question, on a total of 8 pages. Please
check to make sure that you have all the questions.
2. A cumulative normal table is attached to the back
ISE 3414
Midterm Exam #2
Tuesday, June 21st, 2011
Instructions:
1. This exam contains a total of 6 problems for a total of 100 points (plus one bonus
problem) on 8 pages. Please check to make sure that you have all the questions.
2. You are permitted to u
ISE 3414
Midterm Exam #1
Tuesday, June 7th, 2011
Instructions:
1. This exam contains a total of 6 problems on 6 pages, for a total of 100 points. Please
check to make sure that you have all the questions.
2. You are permitted to use one 8.5 x 11 hand-writ
ISE 3414
Midterm Exam #1
Tuesday, June 5th, 2012
Instructions:
1. This exam contains a total of 5 problems on 6 pages, for a total of 100 points. There
is also a 5 point bonus question. Please check to make sure that you have all the
questions.
2. You are