What do you mean by "approaches a continuous distribution"? For a simpler example, P(x)=1/a on (0,a) doesn't approach any probability distribution as a->inf (and it's already continuous for any positive a).

For large n and p>0 the binomial goes into the Gaussian with avg m=n*p and sd= (n*p*(1-p))^(1/2) .

Additionally if n->inf and p->0 but n*p->k>0 it goes to the Poisson distribution.

Is this what you wanted to say?

yes I think what masque de Z said

edit:

so this is part of the lecture notes:



This part of the notes lead me to wonder about the Prob(K = k/sqrt(n)) type distribution

And also I forgot an (n Choose k) up there obv

[X-posted from SMP HW Help Thread, realized this is a better place for it]

Hey Jason & co,

Roommate needs some stats help for work. So here's the question:

He has 300 documents and wants to sample X of them to get a 95% confidence interval with 3% margin of error;. How do we figure out what X is without knowing what the probability of an error in a given doc is?

Let me know with any questions; trying to figure this out.

Thanks Jason,
Mariogs

How would you describe the difference between uncertainty and risk in layman's terms. Are there any nice analogies that you came up with that you use when teaching?

Thoughts on Gödel/Turing/Church and especially Chaitin I suppose from a probabilist's POV?

Even if you let

Xn = ( B(n,p) - np) / Sqrt[np(1-p)]

then the comments in A and B still hold. Notice in A and B he's talking about densities. Xn is discrete so does not have a conventional density function, thus A. Also, for any real x, P(Xn=x) => 0 so B holds.

What is true is that Xn => Z in distribution (or "Law") where Z ~ N(0,1). What this means is that the cumulative distribution functions of Xn converge to the cumulative distribution function of Z.

ie. P(Xn <= x) converges to P(Z <=x) for every real x where the CDF of Z is continuous (which for Z is all x).


See Wiki - Convergence of random variables.

They give an example of random variables which do have density functions and which converge in distribution to a nice uniform distribution but where their density functions do not converge at all.




PairTheBoard

Huh, feels weird a cdf can converge in distribution, but not density. What you say makes sense to me though, ty

The example they give in Wiki involves continuous density functions which oscillate with increasing frequency between the values 0 and 2 on the interval (0,1). They clearly don't converge pointwise but their integrals over any subinterval of (0,1) converge to the length of the subinterval - the oscillations average out in the integral. So their cdf's converge to the uniform distribution on (0,1).

fn = (1 - cos(2pi*nx)* 1_{x in (0,1)}

where 1_{x in (0,1)} is the indicator function on the interval (0,1)
ie.
1_{x in (0,1)} = 1 if 0<x<1 and 0 otherwise.


PairTheBoard

Hands down greatest SMP thread of all time and in the running for all time 2+2 best IMO. The knowledge ITT is excellent but what makes Jason1990 and this thread so bad ass is the occasional poster that decides to step up by saying he doesn't understand the question or thinks they have some brilliant pithy post. He bitch slaps them back in their place more politely than anyone I have ever seen. Thanks for the thread Jason1990 :-)

Let X be a partially ordered set and P and Q be probability measures on X. Suppose that for each principle upset U of X, we have that P(U)<=Q(U). Can you give an example where P is not stochastically smaller than Q?

I don't recall having seen these concepts of "upsets" and "stochastically smaller" before. Looking at Wiki I see "stochastically smaller" defined for random varibles. ie. If A and B are random variables then A is smaller than B means that for every real x, P(A > x) <= P(B > x). Conceptually, A has more probability weight on smaller real numbers relative to B.

However, in your question you don't have any random variables. You have probablity measures on X but no function from X to R defining random variables. So I assume you mean to have a function, say A, from X to R defining two random variables via P and Q, say Ap and Aq. And you then want Ap not smaller than Aq.

It seems to me that you would only expect Ap to be smaller than Aq if the partial ordering on X somehow agreed with the natural ording on R under the map A. So look for a counter example where the map A does not respect the ordering on X. How about letting X = R with the natural ordering, assume you have your P striclty smaller than Q, and let the map A:X --> R be A(x) = -x.

Or assume you have a Q and P on R with Q strictly smaller than P, then let X = R with reverse ordering, and A the identity map.


Or am I missing something?


PairTheBoard

Let (X,F,P) and (X,F,Q) be two probability spaces. P is said to be stochastically smaller than Q if P(U) <= Q(U) for every upset U of X. Other terminology for upset is 'ideal', in case you've heard of that. A subset U of a partially ordered set X is said to be an upset if a in U and a<=b implies b is in U. There is no need to define a random variable for this. Additionally, I don't think the problem can be solved using a totally ordered set for X.

I should also mention that a principle upset is an upset generated by a single element in a partially ordered set.

At least it is easy to find orders with lots of P,Q s.t. P(U)=Q(U)=0 for any principal subset.

e.g. on R with the trivial ordering (x<=y iff x=y), any P,Q that do not charge single points will do.

If you add the requirement that P(U)>0 for all principal subset then it is more difficult.

Thanks--that works. I kept trying to solve the problem using finite partially ordered sets (I'm sure it's possible but I'm not being creative enough I guess).