1. 1 Introduction to Stochastic Models GSLM 54100

2. 2 Outline  course outline course outline  Chapter 1 of the textbook

3. 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

4. 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

5. 5 Examples  Give an outcome, the sample space, and an event of the following experiment  rolling a dice  rolling two dice  flipping coins indefinitely

6. 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?

7. 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

8. 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)

9. 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)

10. 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 )

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

12. 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.

13. 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) = ?

14. 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) =

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

16. 16 Independent Events  events A and B are independent iff P(A  B) = P(A)P(B)  P(A|B) = P(A)

17. 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?

18. 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) = ?

19. 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.

20. 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 )

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

22. 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.