What would P([nU] = n) = P(U = 1) be?
Hi, how to prove that U~uniform[0,1] is equivalent to (i.e. if and only if) [nU]~uniform{0,1,2,...,n-1} for all positive integers n. [] is the floor function. I totally have no idea how to start, so guidance would be greatly appreciated. Also, why is the range from 0 to n-1? If you have [0,1] in the beginning and you multiply by n, shouldn't you have from 0 to n? I am so confused.
What would P([nU] = n) = P(U = 1) be?
I don't have emotions and sometimes that makes me very sad.
geniusacamel (10-14-2016)
Anyway, from your question, it has given me an idea.
[nU] = n-1 if and only if n-1 <= nU < n if and only if (n-1)/n < U < 1. And probability is 1 - n/n + 1/n = 1/n.
[nU] = n-2 if and only if n-2 <= nU < n-1 if and only if (n-2)/n < U < (n-1)/n. Probability is 1/n again.
Continue with n-3, n-4 etc. Is it correct?
Last edited by geniusacamel; 10-14-2016 at 07:47 PM.
So is my above solution good enough or more is needed? Anyway, I need help for a further question:
U_n = [nU]/n. Compute P(U_(n+1) > U_n) and P(U_(n+1) = U_n) for each n in N (natural number) to show that the convergence is not monotone.
I have no idea how to do it at all and I will show you my attempt and I won't be surprised if I get it totally on the wrong track.
So here is my attempt:
[(n+1)U]/(n+1) > [nU]/n when k/(n+1) <= U < k/n , where k is an integer from 1 to n
[(n+1)U]/(n+1) = [nU]/n when U < 1/(n+1) or when U=1
So P(U_(n+1) > U_n) = summation (k/n - k/(n+1)), k from 1 to n = 1/2
P(U_(n+1) = U_n) = 1/(n+1)
Assuming that my probabilities are correct (please let me know if I am wrong), how do I show that the convergence is not monotone?
I think a better way to do this is to show convergence in probability - which is weaker than convergence almost surely.
So, consider a sample space with a Uniform probability measure P. That is, the probability associated with the interval is .
Let be a sequence of intervals of the form for and where , where
,,..., .
Next, define a sequence of random variables on this sample space as for
.
Let and note that is 1 only on the interval such that
,
where is a sequence that converges to 0, .
It follows that for any ,
,
so then it necessarily follows that as .
Well, to be more clear, you compute the intervals like this: m(1)=[0, 1], m(2)=[0, 1/2], m(3)=[1/2, 1], m(4)=[0, 1/3], m(5)=(1/3, 2/3), m(6)=[2/3, 1] - see the pattern.
What I am showing is convergence in probability. It will not converge almost surely. As I said, convergence in probability is weaker than convergence almost surely.
In terms of whether your question regarding monotonicity: Are you being asked to use the Lebesque's Monotone Convergence Theorem???...I guess, I'm just not sure why you asking this.
Well, the problem asked me to find those probabilities and to use them to show that the convergence is not monotone.
Okay, so their is convergence in probability (and not convergence almost surely) - but the entire sequence of intervals m(n) is not (strictly) monotone:
m(1)=[0, 1], m(2)=[0, 1/2], m(3)=[1/2, 1], m(4)=[0, 1/3], m(5)=(1/3, 2/3), m(6)=[2/3, 1], m(7)=[0, 1/4], m(8)=[1/4, 1/2], m(9)=[1/2, 3,4], m(10)=[3/4, 1].
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