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 Mar 27 comment Conditioning on a random variable Multiplying leftmost and rightmost sides by $Pr(\Lambda = \lambda)$ (and dropping the middle term from the equation), we have $$Pr(X \geq 3 |\Lambda = \lambda)Pr(\Lambda = \lambda) = Pr(X \geq 3)$$ Rearranging, I have what you have: $$Pr(X \geq 3) = Pr(X \geq 3 |\Lambda = \lambda)Pr(\Lambda = \lambda)$$ And now we do this for all $\lambda$ Mar 27 comment Conditioning on a random variable Not a problem at all; repetition is always good as long as it clears things up. So I think I see it now. So essentially we're fixing $\Lambda$ and asking ourselves what the desired probability would be for every value of $\lambda$. But we also need to be aware of the probability that $\Lambda$ takes on a certain value. In terms of conditional probability: $$Pr(X \geq 3 |\Lambda = \lambda) = \frac{Pr(X \geq 3, \Lambda = \lambda)}{Pr(\Lambda = \lambda)} = \frac{Pr(X \geq 3) Pr(\Lambda = \lambda)}{Pr(\Lambda = \lambda)}$$ multiplying both sides by $Pr(\Lambda = \lambda$) we have...next post Mar 27 comment Conditioning on a random variable But regarding the $ds$ stuff, so what you're saying is that the probability that $\Lambda$ is on the set (0,5) is $$\frac{(s+ds)-s}{5}$$ and the $s$ cancels out so we're left with $$\frac{ds}{5}$$ But after that I'm lost because you made the variable we're integrating over $x$ and I don't see any $x$'s in the integration. Typo perhaps? Mar 27 comment Conditioning on a random variable Sorry, deleated comment to fix $LaTeX{}$ issues, so they're out of order now, but anyway. So what you're saying is that we multiply the density and mass functions together because by the definition of conditional probability: $$Pr(X = k | \Lambda = \lambda) = \frac{Pr(X = k , \Lambda = \lambda)}{Pr(\Lambda = \lambda)}$$ Mar 27 asked Conditioning on a random variable Mar 24 comment Expectation of a Poisson Process Ah, now it's clear. Exactly what I wanted; once you said it wasn't really recursion, then it ecame obvious to just factor out an $E[W]$ and solve. Mar 24 revised Expectation of a Poisson Process added 213 characters in body Mar 24 comment Expectation of a Poisson Process Yes, like I say above, I'm confused about what to do with this recursive integral. I'm gonna go to bed; need to be up in 3 hours. But I will look at this again tomorrow (or technically later today) Mar 24 comment Probability question with interarrival times Ok, thanks. Could you take a look at my question? It is somewhat similar to this one, but the approach I took is different. math.stackexchange.com/questions/724228/… Mar 24 awarded Excavator Mar 24 revised Probability question with interarrival times corrected grammar Mar 24 comment Expectation of a Poisson Process meh, not entirely; it confuses me more I think. Mar 24 comment Probability question with interarrival times What does $D_i$ represent in therms of the problem? Is it the arrival of car $i$ ? Mar 24 suggested approved edit on Probability question with interarrival times Mar 24 revised Expectation of a Poisson Process added 5 characters in body Mar 24 asked Expectation of a Poisson Process Mar 22 accepted Conditional expectation of an exponential random variable Mar 22 asked Conditional expectation of an exponential random variable Mar 10 comment Winning a restricted game of Nim? Ah...that's what I was missing. I thought we were just xor-ing the size of the piles (which is the Grundy value of the starting positions of each pile anyway). I didn't realize I had to find new Grundy numbers. Mar 10 accepted Winning a restricted game of Nim?