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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?