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Introduction to Stochastic ModelsIntroduction to Stochastic ModelsGSLM 54100GSLM 54100

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OutlineOutline

course outline

Chapter 1 of the textbook

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Some Standard TermsSome Standard Terms

experiment: the collection of tasks to get raw data (samples, observations) in studying a given (random, stochastic) phenomenon

outcome: a sample data got from an experiment

sample space: the collection of all outcomes

event: a collection of some outcomes

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Relationship Between Relationship Between Outcome, Event, and Sample SpaceOutcome, Event, and Sample Space

sample space : the universal set

outcome: an element of

event: a subset of

new events from , , and ()c of events

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ExamplesExamples

Give an outcome, the sample space, and an event of the following experiment

rolling a dice

rolling two dice

flipping coins indefinitely

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More Examples on EventsMore Examples on Events

assign meaning to an event

what is the event of {2, 4, 6} in rolling a dice?

use compact ways to represent an event

how to represent

the event that the sum of the two dice is greater than or equal to 5 in rolling two dice?

the event that the number of heads is no less than the number of tails in infinite coin flipping?

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Probabilities Defined on EventsProbabilities Defined on Events

the probability P() is a function defined on event that has the following properties: (a) P(A) 0 for any A (b) If Ai’s are mutually exclusive subsets of , i.e.,

Ai and AiAj = for i j, then P(A1A2 ...) = P(A1) + P(A2) + ...

(c) P() = 1

these properties being sufficient to deduce all other results

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Derivation of …Derivation of …

P(Ac) = 1 - P(A)

P() = 0

if A B, then P(A) P(B)

0 P(A) 1

P(AB) = P(A) + P(B) - P(AB)

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Example 1.3Example 1.3

tossing two coins, equally likely to have any of the four outcomes to appear

find P( either the first coin or the second coin is a head) by listing out all outcomes

by P(AB) = P(A) + P(B) - P(AB)

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Some More Results Some More Results

P(E1E2… En)

= i P(Ei) - i<j P(EiEj) + i<j<k P(EiEjEk)

i<j<k<l P(EiEjEkEl) + …

+ (1)n+1P(E1E2…En)

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Conditional ProbabilitiesConditional Probabilities

the probability of A given B (has occurred)

( )( | ) = , ( ) > 0

( )

P ABP A B P B

P B

A B

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Example 1.5Example 1.5

a family of two kids, each being equally likely to be a boy or a girl

Given that the family has at least a boy, what the probability that the family has two boys?

Is this the way: given that there is at least a boy, there is half and half chance for the other being a boy. Therefore, the conditional probability is 0.5.

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Example 1.7Example 1.7

an urn of 7 black balls and 5 white balls

two balls randomly drawn without replacement

P(both balls are black) = ?

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Example 1.7Example 1.7

two ways to solve

by counting:

by conditional probability:

P(two balls are black)

= P(first ball is black)P(two balls are black|first ball is black)

= P(first ball is black) P(the second ball is black|first ball is black)

=

7 67! 5!7 5

1 22 0 2!5! 0!5!12 12! 12 112 2!10! 1 2

(1) 7

22 =

C C

C

7 6 7

12 11 22

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Example 1.8Example 1.8

three men mixed their hats and randomly picked one

find P(none picked back his hat)

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Independent Events Independent Events

events A and B are independent iff

P(AB) = P(A)P(B) P(A|B) = P(A)

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Example 1.9Example 1.9

three events related to rolling two fair dice

E1: the sum = 6

E2: the sum = 7

F: the first die lands 4

Are E1 and F independent?

Are E2 and F independent?

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An Example Similar to Example 1.10:An Example Similar to Example 1.10:Pairwise Independence Does Not Imply IndependencePairwise Independence Does Not Imply Independence

three events for flipping two fair coins

A: the first coin lands head

B: the second coin lands head

C: the two flips give the same result

P(A) = ? P(B) = ? P(C) = ?

P(A|B) = ? P(A|C) = ? P(ABC) = ?

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Example 1.11Example 1.11

This is a very interesting example. We will discuss it again after we have gone over indicators and the discrete uniform distribution.

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Baye’s FormulaBaye’s Formula

P(A) = P(A|B)P(B) + P(A|BC)P(BC)

one of the most important equation of the course

a generalization:

for B1B2… Bn = , Bi Bj = for i j

P(A) = i P(A|Bi)P(Bi)

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Worksheet #3

Exercises #3, 4, 5, 6

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Assignment #1Assignment #1

Here are some simple problems in Chapter 1 of the textbook: Ex 1.1, Ex 1.18, Ex 1.20, Ex 1.26, Ex 1.34.