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Kvisoft Flipbook Maker Pro 3.6.3 Full Version 4SHARED kvisoft flip book maker, kvisoft flipbook maker pro, kvisoft flip book maker pro 3.6.5.0, .Q:

Conditional expectation in Poisson process

Suppose $X_1,X_2,\cdots$ is a sequence of independent Poisson random variable with parameters $ \lambda,\mu,
u$ respectively. Also suppose $Y_n=\min\{X_1,\cdots, X_n\}$ and $Z_n=\max\{X_1,\cdots, X_n\}$.

What is the conditional distribution of $Y_n|Y_1,\cdots, Y_{n-1}, Z_1,\cdots, Z_n$?

A:

Conditional on $Y_1,\dots,Y_{n-1},Z_1,\dots,Z_n$, the variables $Y_n,X_1,\dots,X_n$ are independent. Using memorylessness of the Poisson process, we can then find $P(Y_n=k|Y_1,\dots,Y_{n-1},Z_1,\dots,Z_n)=e^{ -\lambda}\frac{\lambda^k}{k!}$, for all $k\geq1$.
Edit: In case $Y_1,\dots,Y_{n-1},Z_1,\dots,Z_n$ are all equal to some $k$, then $Y_n=k$. Thus, the distribution of $Y_n|Y_1,\dots,Y_{n-1},Z_1,\dots,Z_n$ is the mixture of these two distributions, i.e. $$P(Y_n=k|Y_1,\dots,Y_{n-1},Z_1,\dots,Z_n)=P(Y_n=k|Y

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