1 introduction to stochastic models gslm 54100. 2 outline course outline course outline chapter 1...
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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)