Basic random variable question - lim of min of a set of RV's
Maths and statistics discussion, revision, exam and homework help.
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Basic random variable question - lim of min of a set of RV'sLet the independent, identically distributed random variables
have common distribution function 
.
Denote
Question: When the distribution of
is uniform over the interval ![[0,1]](http://www.thestudentroom.co.uk/latexrender/pictures/4c/4c9556ccb7d639c92940e3db1eca4b07.png "[0,1]")
, find the limit 
for 
.
So far, this is what I have:
![P\{Z\leq x \} = 1-[1-F(x)]^n](http://www.thestudentroom.co.uk/latexrender/pictures/00/007e17dac982d2fb589f815fa42b3280.png "P\{Z\leq x \} = 1-[1-F(x)]^n")
![\therefore P\{nZ\leq x \} = P\{Z\leq \frac{x}{n} \} = 1-[1-F(\frac{x}{n})]^n = 1-\left(1-\frac{x}{n}\right)^n \to \ ?](http://www.thestudentroom.co.uk/latexrender/pictures/54/542d4cf8933ac25e3fdf2a97e5bb5227.png "\therefore P\{nZ\leq x \} = P\{Z\leq \frac{x}{n} \} = 1-[1-F(\frac{x}{n})]^n = 1-\left(1-\frac{x}{n}\right)^n \to \ ?")
Well basically I'm just double checking this is correct. If it is, then would the final term tend to 1? If so, why exactly? -
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Re: Basic random variable question - lim of min of a set of RV'sYou may find this helpful:(Original post by wanderlust.xx)
...
http://en.wikipedia.org/wiki/Exponential_functionLast edited by ghostwalker; 17-05-2012 at 11:00. -
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Re: Basic random variable question - lim of min of a set of RV'sAh yes of course. Cheers.(Original post by ghostwalker)
You may find this helpful:
http://en.wikipedia.org/wiki/Exponential_function
