Question

Text Book: Introduction to Probability Models 11th edition, Sheldon M. Ross: Chapter 5, Question 42(b)

Can you please explain how the expression in b was derived and which theorem (or provide page # of the theorem) the question is referring to? I see the solution but I don't understand it. (like why T1+T2<=1)

Thank you!

Answer 1

See this is all about a little bit on intuition N(t) is number of events that occurred till the time t. now there are are two three facts about poisson process that we need to know .A poisson process has indepenpendent increments...now to get this into intuition this implies that for any two disjoint interval of times the distribution of the number of events occuring in the first interval and the distribution of number of events occurring in the other disjoint interval are independent of each other and infact here its identical also...this means say number of events happening in interval [2,3] is independent of the no. of events occurring in say [4,5] or anyother interval which doesnt overlap with [2,3] i.e is disjoint to [2,3].

Now comes the second fact that in a poisson processwith rate lamda,the time taken for 1st event to occur(denote by T₁ follows exponential(lamda) and the also that the time taken for second event to occur given T₁ also follows exponential(lamda) because of the first fact that poisson process that independent increment.

So thats is why in the first problem ie 42 a) S₄ can be written as S₄=T₁+T₂+T₃+T₄ where T_i = time taken for 1 event to occur after the (i-1)th event has occured.Which makes them independent of each other. and they all identically follow exponential(lamda). and thats why epxectation comes out as:

now coming to 42 b) E(S₄ | N(1)=2) ... first we need to understand what this condition changes...

as we know N(t) is no. of events occurred till time t. and hence N(1)=2 means that till time t=1 , exaclty 2 events have occurred so you said you didnt understand why in the solution is T₁+T₂ <=1, see as we have defined T_i's in the way that we did... this is obvious because remember T₁  is time taken for first to occur and T₂ is time taken after the ocurrence of 1st event, for the second event to occur so obviosly total time taken for 2 events to occur will be?? T₁+T₂ .Good.Simple right?now as the condition is  N(1)=2 that says till time 1 exactly two events have occur (here 'till' is the catch...it doesnt say at exactly what time 2 events have occurred....it just says that till time=1 i.e it might happen extact time was 0.3 at which two events had taken place.) so remember this...so as T₁+T₂ is our exact time at which two events have ocuurred this along with the condition mean that T₁+T₂<=1.So now we have understood what it actually means

Now coming to the solving E(S₄ | N(1)=2) ....now we know that as till time 1 only two events have occurred so for S4  

we need two more events to occur after time=1 but we know they are independent of the what and how many occurred till time=1 right? so so the answer is 1+(expected time taken for 2 more events)= which is our required answer.

I went through every detail and tried to give a proper insight as to whats happening and why is it that way. thats took the help of the first problem 42 a)....sorry if i was a bit slow in the process...

Cheers.