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1、Conditionality and stopping times in probability,Mark Osegard, Ben Speidel, Megan Silberhorn, and Dickens Nyabuti,Conditional Expectation,Conditional Probability,Discrete: Conditional Probability Mass Function,Continuous: Conditional Probability Density Function,Conditional Expectation,Discrete:,Con
2、tinuous:,Note:,of y. We write this as,is a function,i.e.,(Conditional Expectation Function),Theorem:,Clearly, when Y is discrete,When Y is continuous,Proof: Continuous Case,Recall, if X,Y are jointly continuous with joint pdf,Define:,and,Note:,Continuous Case Cont.,(Fubinis Theorem),So,Therefore, co
3、ncluding,Summary:,When Y is discrete,When Y is continuous,Conditional Variance,Definition,Proof,Note as well ,adding,g,Stopping times,Stopping Times,DefinitionApplication to ProbabilityApplications of Stopping Times to other formulas,Stopping Times,Basic Definition:A Stopping Time for a process does
4、 exactly that, it tells the process when to stop.Ex) while ( x != 4 ) The stopping time for this code fragment would be the instance where x does equal 4.,Stopping times in Sequences,Define:Suppose we have a sequence of Random Variables (all independent of each other) Our sequence then would be:,Sto
5、pping Times: A Discrete Case,From our previous slide we have the sequence:A discrete Random Variable N is a stopping time for this sequence if : N = n Where n is independent of all following items in the sequence,Independence,Summarizing the idea of stopping times with Random Variables we see that t
6、he decision made to stop the sequence at Random Variable N depends solely on the values of the sequence Because of this, we then can see that N is independent of all remaining values,Applications of Stopping Times,Does Stopping Times affect expectation?No!Consider this statement:This formula, the fo
7、rmula used for Conditional Expectation does remain unchanged,Applying Stopping Times,For an example of how to use stopping times to solve a problem, we will now introduce to you Walds Equation,Walds Equation,Proposition,If X1, X2, X3, are independent identically distributed (iid) random variables ha
8、ving a finite expectation EX, and N is a stopping time for the sequence having finite expectation EN, then:,Walds Proof,Let N1 = N represent the stopping time for the sequence X1, X2, , XN1Let N2 = the stopping time for the sequence X(N1+1) , X(N1+2), , X(N1+N2)Let N3 = the stopping time for the seq
9、uence X(N1+N2+1) , X(N1+N2 +2), , X(N1+N2+N3) ,Walds Proof .,We can now define the sequence of stopping times as where Ni clearly represents,and see the sequence is iid,Walds Proof,If we define a sequence Si as,where,Note: Si are iid,Walds Proof,which are iid because the Xis are.,Walds Proof,By the
10、Strong Law of Large Numbers,Walds Proof,Also,Concluding,So as we let,Which can be manipulated into our preposition:,Miners Problem,Sample Conditional and Stopping times in probability problem,The problem,A miner is trapped in a room containing three doors. Door one leads to a tunnel that returns to
11、the same room after 4 days; door two leads to a tunnel that returns to the same room after 7 days; door three leads to freedom after a 3 day journey. If the miner is at all times equally likely to choose any of the doors, find the expected value and the variance of the time it takes the miner to become free,Expected Value,Using Walds Equation:,Continue .,Continue .,Expected value Conclusion,Variance,Continue.,Continue.,Continue.,Thus far,| | | | ?,Continue.,Continue.,Continue.,In conclusion,| | | |,Thanks to:,Dr. Steve DeckelmanDr. Ayub HossainWho helped make this a success!,
