April 2013 Exam - 8(iv)

Discussion in 'CT4' started by sfischer, Apr 14, 2014.

  1. sfischer

    sfischer Member

    In the April 2013 Exam que 8(iv), did we come to the answer by first principles or via the Kolmogorov equations. I cant see how the Kolmogorov equations came to the conclusion. If by first principles, are we saying the probability of holding in state U from 0 to t is the integral in s of Puu(s) from 0 to t?

    Thinking about it further - aren't we simply being asked the probability that it holds in state U for 1.5 hours and then instantly goes to state R. In which case the answer would be 1/40*(exp(-1.5*9/40)) however the answer is 1/9*(1-exp(-1.5*9/40))?
     
    Last edited by a moderator: Apr 14, 2014
    sfischer, Apr 14, 2014
    #1
  2. John Potter

    John Potter ActEd Tutor Staff Member

    You could work out the probability that the player is sent off during the match directly using the Kolmogorov differential equations, but this would not be right because it includes players who end up with a red card having previously been booked. To incorporate the extra condition “without previously having been cautioned”, we need to use an integral.

    The integral is based upon your logic below

    You're making the jump just at time 1.5, when the jump could go on at any time t between 0 and 1.5. Have another go, build an integral between 0 and 1.5 and use your logic - stay in State U from 0 until t, jump from U to R. THis could happen for any time t between 0 and 1.5.

    John
     
    John Potter, Apr 14, 2014
    #2
  3. sfischer

    sfischer Member

    Ok - I misread the question (going to need to work on that in the next 2 weeks). I read it that the jump happened at 1.5 hours but your right - it can happen at any time from 0 to 1.5. Thanks for your help.
     
    sfischer, Apr 14, 2014
    #3
  4. John Potter

    John Potter ActEd Tutor Staff Member

    Just as a little tip on this one... jumps cannot happen at exact times, in the same way a continuous RV can not equal an exact value. So, perhaps, this in itself could maybe have helped you realise that the question needed reading again?

    Good luck!
    John
     
    John Potter, Apr 16, 2014
    #4
  5. sfischer

    sfischer Member

    Thanks for the tip. I can use all I can get.

    Also on a separate note, do you think we will need to remember the formulas for the integrated Kolmogorov equations - time-inhomogenenous and time-homogeneous? I haven't seen an example in past papers as yet.
     
    sfischer, Apr 16, 2014
    #5

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