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Sept 2021 Q8iv

L

Laura

Very Active Member
Hi all,

Could you clarify why if X_t is a random variable denoting the amount of time spent offline over the period [0, t], given that the customer is offline at time 0, then the expected value of X_t is given by: E(X_t) = INT (0, t): P_OFF_OFF(s) ds ?
Is there a formula i can locate in the notes for obtaining E(X_t)?

Thanks in advance for your help!
 
Hello

This is a tricky one! Unfortunately, I don't believe that there is a formula in the Core Reading for this. To see why it makes sense think about breaking up the time interval 0 to t into steps of size h where it is only possible to occupy one state in each interval of this length. In this discrete version, the expected amount of time spent in the off state is sum(over k) P(being in off state in the kth interval) * h) ie:

P(being in off in interval 0 to h) * h + p(being in off in interval h to 2h) * h ... + p(being in off in interval t-h to t) * h

This is:

p_OFF_OFF(0) * h + P_OFF_OFF(h) * h + P_OFF_OFF(2h)*h + ... + P_OFF_OFF((t-h)) * h

As we let h tend to 0, the continuous version of this is int(0,t) p_off_off(s) ds

Hope this helps!

Andy
 
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