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I have the following generating function for stochastic process

$$\sum_n z^n p_n=\left[1-\frac{(1-z)}{\Lambda(t)}\right]^{n_0},$$

and I want to extract the probability $p_0(t)$ but I am confused how I could do this. So far I've thought of

$$p_0(t)=\sum_n z^n p_n \delta_{n, 0},$$

to extract it but that doesn't seem to help solve this. Any help would be appreciated.

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  • $\begingroup$ what happens if you evaluate at $z=0$? $\endgroup$
    – Phicar
    Jan 14, 2022 at 14:01
  • $\begingroup$ @Phicar Wow! I'm embarrassed. I knew it was something simple but my brain just wouldn't let me. Thank you! $\endgroup$
    – adammoyle
    Jan 14, 2022 at 14:04
  • $\begingroup$ No worries, happens to me all the time. Welcome here! $\endgroup$
    – Phicar
    Jan 14, 2022 at 14:08
  • $\begingroup$ @RobPratt Ok, I have done so Rob. Thanks for caring. $\endgroup$
    – Phicar
    Jan 14, 2022 at 20:49
  • $\begingroup$ Yep, just gave you a +1. $\endgroup$
    – RobPratt
    Jan 14, 2022 at 20:50

1 Answer 1

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Evaluate at $z=0$ in your first expression. Notice that for $n>0$, one has $0^n=0$ and the only term remaining is the constant term $p_0$.

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