Recent content by infk

1. There are ##\binom{n}{2}## ways to do that. 2. There should be in total ##n## recordings of the only two unique values. If the first one occurs once, the other one must occur ##n-1## times, or the first one occurs twice, then the other one occurs ##n-2## times, and so on and so forth...

If we have ##x_1, \ldots , x_n##, all distinct values, and then sample from this with replacement and thus obtain a bootstrap sample ##x^{\star}_1, \ldots , x^{\star}_n##, what is the probability that the bootstrap sample has only two unique values?My attempt at a solution: there are...

Hi and thanks for the response. What I meant was of course that ##h^{-1}_1 \circ f = g \circ h_0##, it seems though that you did not notice my typo. Also, could you use subscript notation, it is not clear which of ##h_1## and ##h_2## you mean. Cheers

Hey, Let ##(f,g) \in B^A## where ##A## and ##B## are non-empty sets, ##B^A## denotes the set of bijective functions between ##A## and ##B##. We assume that there exists ##h_0: A \rightarrow A## and ##h_1: B \rightarrow B## such that ##f = h_1 \circ g \circ h_0 ##. This implies that ##g =...

Ok, have a look here then: http://mathworld.wolfram.com/DisjointUnion.html

So there is not a single person on the board who knows about this? It is from a first course in discrete mathematics for 2nd year students.

I thought this was standard notation in (discrete) mathematics, but anyway, ##A \leftrightarrow B## means that there exists a bijection between ##A## and ##B##.

##A \sqcup B## is the disjoint union of ##A## and ##B##, defined by: ##A \sqcup B = \{(x,0):x \in A \} \cup \{(y,1):y \in B \}##

Hi, So I am trying to show the following: ##(A \cup B)\sqcup(A \cap B) \leftrightarrow A \sqcup B## The proof that I am trying to understand starts with: ##A \leftrightarrow (A \backslash B) \sqcup (A\cup B) \qquad (1)##, and ##A \cup B \leftrightarrow (A\backslash B)\sqcup B \qquad...

Same question but we pick another ##T##: ##T = (\text{min}(\mathbf{X}),\text{max}(\mathbf{X}))##. In this case we should get the same probability ##P(T(X)=t)## as before, namely ##P(T(\mathbf{X}) = t)= n!P(\mathbf{X} = X)##, for consider the case: ##P(T = 1,X_2,3)## this is the sum of the...

I see what you mean. But since ##P(T(\mathbf{X}) = (1,2)) = P(\mathbf{X} = (1,2)) + P(\mathbf{X} = (2,1)) = 2*P(\mathbf{X} = (1,2))##, shouldn't it rather be: P(T(\mathbf{X}) = t)= n!P(\mathbf{X} = X)? This way, the probability ##P(\mathbf{X} = X|T(\mathbf{x}) = t)## is equal to...

Hello Let's say we have some continuous i.i.d random variables X_1, \ldots X_n from a known distribution with some parameter \theta We then place them in ascending order X_{(1)}, \ldots X_{(n)} such that X_{(i)}, < X_{(i+1)}. We call this operation T(\mathbf{X}) where \mathbf{X} is our...

Sorry to bump this thread,but I still haven't figured it out. Does anyone know how to find it?

Homework Statement Let {U_k}_{k \in \mathbb{N}} be i.i.d from the uniform (0,1) distribution. I need a formula for the cumulative distribution function of X_n, defined as X_n := n* \min(U_1, \ldots ,U_n) Also some advice for X_n := \sqrt{n}* \min(U_1, \ldots ,U_n) would be appreciated. * is...

Homework Statement So here is a "proof" from my measre theory class that I don't really understand. Be nice with me, this is the first time I am learning to "prove" things. Show that a connected open set Ω (\mathbb{R}^d, I suppose) is a countable union of open, disjoint rectangles if and...