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Old 04-27-2017, 05:53 AM   #1
boganomics
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Mathematical Probability Question

Hi all, I posted this in SMP (under new account, couldn't remember my pw) and dunno if this is a more relevant place to post. I tried for a bit and have no idea how to approach it:

u and v are i.i.d sampled from the uniform distribution [0,1]

if s^2 = (2u-1)^2 + (2v-1)^2 < 1, then x is sampled from the standard normal distribution. Otherwise, x = (2u-1) * sqrt( -2log(s)/s)

Find the distribution of x


I know general CDF method etc. I also think there's a bug because some values of u and v will result in a negative square root, but it'll still help to have some general direction in how to approach this
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Old 04-27-2017, 03:54 PM   #2
TomCowley
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Re: Mathematical Probability Question

That distribution doesn't make sense for the reasons you described. If you look at https://en.wikipedia.org/wiki/Box%E2...ller_transform under the polar form (2 * uniform[0,1] - 1 is uniform [-1,1] obviously), the last part of the calculation is identical to the "otherwise" in yours.... but that's intended for cases where your s^2<1 inequality holds. Are you sure you aren't misreading the question and adding an "otherwise" that doesn't belong and that's actually supposed to be the definition of x as a standard normal deviate?
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Old 04-27-2017, 06:26 PM   #3
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Re: Mathematical Probability Question

Hi Tom,

I'm 100% not misreading so it's obviously a mistake, I can see another student has asked the lecturer for clarification, hopefully he replies soon. Thanks a lot for the link, it definitely helps (and is interesting reading!). I'll attempt the question when I know what the correct form is and maybe come back here if it's particularly trying
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Old 04-28-2017, 04:05 AM   #4
boganomics
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Re: Mathematical Probability Question

Lecturer confirmed, it was a mistake, should be a >=1 for the condition on s^2
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