mod03lec17 - Conditional arrival density and order statistics - Part 2

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Conditional joint density of arrival epochs, Conditional density of interarrival times, marginal conditional density of interarrival times, uniform order statistics
  1. NPTEL-NOC IITM
  2. IID uniform
  3. interarrival times
  4. marginal distribution of epochs
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@20:48 I found the below line of reasoning useful.
1) We have already shown P(S_1 > x | N(t) = n) = (t-x)^n/t^n. We got this as We have to distribute n uniforms in time [0, t-x] instead of [0, t] now.
2) S_1 = X_1 P(X_1 > x | N(t) = n) = (t-x)^n/t^n 
3) X_i's are independent and one can't distinguish X_1 from the rest. P(X_i > x | N(t) = n) = P(X_1 > x | N(t) = n) i in [n]
4) P(X_i > x | N(t) = n) = (t-x)^n/t^n

RahulMadhavan