TogoPogo Posted September 8, 2012 Report Share Posted September 8, 2012 I came across this IB question on a proof for Poisson distribution and I'm stuck as to how to approach this. I tried expanding the right side but I'm stumped as to what to do with the sigma: Link to post Share on other sites More sharing options...
aldld Posted September 8, 2012 Report Share Posted September 8, 2012 (edited) Your image is broken.ETA: If it involves sigma (as in sum, not standard deviation) I'm going to go out on a limb without even seeing the problem and suggest trying induction . Edited September 8, 2012 by aldld Link to post Share on other sites More sharing options...
pravzcool Posted September 8, 2012 Report Share Posted September 8, 2012 Are you sure that is an IB Question ? Link to post Share on other sites More sharing options...
TogoPogo Posted September 8, 2012 Author Report Share Posted September 8, 2012 Sorry, here's the URL: http://i.imgur.com/RLcFL.pngPart B requires induction so I'm not sure what to do for A Link to post Share on other sites More sharing options...
aldld Posted September 8, 2012 Report Share Posted September 8, 2012 (edited) Sorry, here's the URL: http://i.imgur.com/RLcFL.pngPart B requires induction so I'm not sure what to do for AYour URL is still not working for me. Try reuploading it.Edit: It's working now.For part a, what you really wish to show is that (should be lambda + mu, LaTeX is a bit wonky on this site)For the right hand side, try expanding out the using the binomial theorem. For the left hand side, use the given identity and substitute the probability mass functions for X and Y and see if you can manipulate it to look like your expression for the right hand side.(Hint: )Let me know if you have any further questions Edited September 8, 2012 by aldld Link to post Share on other sites More sharing options...
TogoPogo Posted September 8, 2012 Author Report Share Posted September 8, 2012 Ah the hint helped a lot! We didn't learn that in class :$ Any hints for b? Is Ur supposed to be a random variable in the sense that P(X=Ur)? Thanks! Link to post Share on other sites More sharing options...
aldld Posted September 8, 2012 Report Share Posted September 8, 2012 (edited) The base case, n = 1, is trivially true.For the inductive step, show that if the random variable U1 + ... + Un is Poisson with mean nm, then U1 + ... + Un + U(n+1) is also Poisson with mean nm + m = (n+1)m, using part a.Is Ur supposed to be a random variable in the sense that P(X=Ur)?Ur is a random variable and you obtain a new random variable by considering U1 + ... Un, there is no X in part b, so really what you want to consider is P(U1 + ... + Un = k). However you don't need to calculate this explicitly, though you could if you wanted. Edited September 8, 2012 by aldld Link to post Share on other sites More sharing options...
TogoPogo Posted September 8, 2012 Author Report Share Posted September 8, 2012 Sorry, but how do I incorporate part a into the induction? Induction was never really my thing haha. I understand the process of induction but I don't see how part a comes into play... Link to post Share on other sites More sharing options...
aldld Posted September 8, 2012 Report Share Posted September 8, 2012 By the inductive hypothesis, U1 + ... + Un is a Poisson random variable with mean nm. U(n+1) is also a Poisson random variable with mean m. If you add U1 + ... + Un and U(n+1) together, what does part a say about (U1 + ... + Un) + U(n+1)? Link to post Share on other sites More sharing options...
LeonieIB Posted September 8, 2012 Report Share Posted September 8, 2012 Sorry for bumping in - I'm new but just wondering...I can ask for some homework help on these forums? Link to post Share on other sites More sharing options...
The Economist Posted September 8, 2012 Report Share Posted September 8, 2012 Sorry for bumping in - I'm new but just wondering...I can ask for some homework help on these forums? You can post your math questions here: Automatic generated messageThis topic has been closed by a moderator.Reason: Question answered.If you disagree with this action, please report this post and a moderator or administrator will reconsider it.Kind regards,IB Survival Staff Link to post Share on other sites More sharing options...
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