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1 Introduction to Stochastic Models GSLM 54100

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2 Outline course outline course outline Chapter 1 of the textbook

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3 Some 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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4 Relationship Between Outcome, 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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5 Examples 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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6 More 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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7 Probabilities 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 A i ’s are mutually exclusive subsets of , i.e., A i and A i A j = for i j, then P(A 1 A 2 ...) = P(A 1 ) + P(A 2 ) +... (c) P( ) = 1 these properties being sufficient to deduce all other results

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8 Derivation of … P(A c ) = 1 - P(A) P( ) = 0 if A B, then P(A) P(B) 0 P(A) 1 P(A B) = P(A) + P(B) - P(A B)

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9 Example 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(A B) = P(A) + P(B) - P(A B)

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10 Some More Results P(E 1 E 2 … E n ) = i P(E i ) - i<j P(E i E j ) + i<j<k P(E i E j E k ) i<j<k<l P(E i E j E k E l ) + … + ( 1) n+1 P(E 1 E 2 …E n )

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11 Conditional Probabilities the probability of A given B (has occurred) A B

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12 Example 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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13 Example 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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14 Example 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) =

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15 Example 1.8 three men mixed their hats and randomly picked one find P(none picked back his hat)

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16 Independent Events events A and B are independent iff P(A B) = P(A)P(B) P(A|B) = P(A)

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17 Example 1.9 three events related to rolling two fair dice E 1 : the sum = 6 E 2 : the sum = 7 F: the first die lands 4 Are E 1 and F independent? Are E 2 and F independent?

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18 An Example Similar to Example 1.10: Pairwise 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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19 Example 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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20 Baye’s Formula P(A) = P(A|B)P(B) + P(A|B C )P(B C ) one of the most important equation of the course a generalization: for B 1 B 2 … B n = , B i B j = for i j P(A) = i P(A|B i )P(B i )

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21 Worksheet #3 Worksheet #3 Exercises #3, 4, 5, 6

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22 Assignment #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.

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