[经济学]Conditionality and stopping times in probability.ppt

Conditionality and stopping times in probability Mark Osegard, Ben Speidel, Megan Silberhorn, and Dickens Nyabuti Conditional Expectation Conditional Variance Definition Proof Note as well … …adding Stopping times Stopping Times Definition Application to Probability Applications of Stopping Times to other formulas Stopping Times Basic Definition: A Stopping Time for a process does 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: Stopping 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 the 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 formula 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 Wald’s Equation… Wald’s Equation Proposition Wald’s Proof Wald’s Proof ... Wald’s Proof… Wald’s Proof… Wald’s Proof… Wald’s Proof… Concluding 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 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.