CS723 - Probability and Stochastic Processes

Lecture No. 08

In Previous Lectures • Conditional probability Joint Probability • Total probability • Bayes rule for finite sample spaces

In Previous Lectures Started analysis of simple problems with infinite sample spaces. Random orientation of disk Random placement of foot of a person on a partitioned footpath Random location of a circular disk as it is thrown on a floor which has been marked by either only horizontal lines / horizontal and vertical lines

Trains on a Junction Simplistic mathematical model for complex real-life situation East-bound train arrives every 13 minutes and stops for 3 minutes Next train comes after 10 minutes

Trains on a Junction person magically appears at the platform at a random time If train is present at the platform, he boards it instantly When the train’s departure time arrives, it suddenly vanishes from our eyes

East bound Train A person enter at random time and finds one of the two situations

Probabilities Pr( person has to wait for the Train)= 5/13 Pr( person does not have to wait)= 3/13 Pr( person has to wait for at the most 2 min/ person has to wait) = 2/10 = 1/5

Two Trains East-bound train arrives every 13 minutes and stops for 3 minutes South-bound arrives every 12 minutes and stops for 2 minutes Two people enters at random times to board the two trains The joint state represents the sample space of their observations

Catching of Two Trains

Pr( none of the commuters have to wait) = 6/156

A Local Bus Stop A boy and a girl always ride the 8:10 bus from a local bus stop They arrive at the bus stop at a random time between 8:00 and 8:10 The bus suddenly appears and takes them away While standing at the bus stop, they have an opportunity see each other

A Local Bus Stop

Random Line Partition A unit length line [0,1] is divided into three parts by two random cuts The two cuts can be at the same point but probability of this event is zero One or both cut points could land at end point but with probability zero The three segments are non-zero length Many interesting events in terms of the lengths of three segments

Left Seg. is Smallest x < 1/3 is vertically hatched y > 2x is horizontally hatched 1- y > x is filled with yellow

Left Seg. is Longest x > 1/3 is vertically hatched y < 2x is horizontally hatched 1- y < x is filled with yellow

Longest Part > 0.75

Shortest Part < 0.25