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PGF of a conditional random variable


New member
May 31, 2012
I have a queueing system.
The probability Generating Function of the number of packets in the queue (queue length) is given by
\(\displaystyle Q_G(z)=\frac{e^{\lambda T(z-1)}(1-z^{-1})(1-\lambda T)}{1-z^{-1}e^{\lambda T(z-1)}}\).

I need to find the PGF of a conditional quantity.
\(\displaystyle X=(Q_G|Q_G>0)\)
i.e. to say in words "what is the PGF of queue length given that the queue length is greater than zero".

For a normal random variable X whose Distribution is known this can be defined as
\(\displaystyle F(x|M)=\frac{{P\{X\leq x,M}\}}{P(M)}\)

I am not sure how to proceed whem pmf is not known.