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September 10th, 2014, 05:11 AM   #1
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Question Poisson riding the bike

Hello,

I hope you can help me. I have been assigned this problem:

When cycling home at night, I notice that sometimes my rear light is switched off when I arrive home. Presumably the switch is loose and can flip from on to off or back again when I go over bumps. I suppose that the number n of flippings per trip has a Poisson distribution e^{-lambda}lambda^{n}/n!. If the probability that the light is still on when I arrive home is p, find lambda.

I tried to solve the problem in this way:

p = P(0) + P(2) + P(4) + .... = e^{-lambda} Sum(k=0; infinity) of lambda^{2k}/(2k)!

Under the assumption that for an even number of flippings the light results to be on... Do you think this is correct? And then I thought that the Sum could be a Taylor expansion of something which is related to e^{lambda}, but I cannot find the relation...

Any suggestion?

Thanks!
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September 11th, 2014, 06:14 AM   #2
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Sorry, there is an evident mistake, the Sum starts from k = 1... Hope to get some hint! I don't even know if it's the right way to start for the solution.
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September 11th, 2014, 08:04 AM   #3
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Quote:
Originally Posted by Cartesio View Post
the Sum starts from k = 1...
No, you need P(0) which corresponds to $k=0$, so I think you had it correct initially. Moreover, everything you have said is correct.

All you need to think about is
$\cosh x = \frac12 ( \mathbb{e}^x + \mathbb{e}^{-x} )$

Or, to put it another way, your series looks like the terms or $e^x$ in even powers of $x$.

Last edited by v8archie; September 11th, 2014 at 08:06 AM.
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