0
votes
1answer
44 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
0
votes
1answer
24 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
2
votes
1answer
57 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
1
vote
1answer
56 views

Which Queue to Join at the Super Market

Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory ...
0
votes
0answers
17 views

Decreasing step size in SA algorithms

I don't understand why deceasing step sizes implies "implicit" averaging of the noise in stochastic approximation algorithms
2
votes
0answers
88 views

Calculating the maximum of a function

How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$ given ...
5
votes
2answers
109 views

Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...
0
votes
0answers
20 views

Randomized optimization: confidence bounds

I am not sure whether the question is appropriate for MSE, so please feel free advise another website in the network. Suppose I have a Lipschitz continuous function $f:X\to \Bbb R$ where $X$ is a ...
0
votes
1answer
77 views

Bounded logarithmic function

I am trying to find any function that it grows logarithmically up to a certain point, and after that point it remains constant. Can anyone help me with that
0
votes
1answer
40 views

Theory of Moments: Notation

I try to read some papers about Moment Matrices/Optimization over polynomials, but I have some troubles with the following notation: Let $P(V)$ be a power set of some $V=\{1,2,...,n\}$, how does a ...
2
votes
0answers
81 views

Learn about reproducing kernel Hilbert spaces?

Why are reproducing kernel Hilbert spaces an important topic to learn? What is possibly achievable with that theory that is not reachable with just standard Hilbert space theory?
1
vote
1answer
148 views

Goofy problem: Optimal bet with nearly no knowledge

A year or so back, on the verge of falling asleep, I thought up this question: You have come to me ready to gamble. I have two envelopes on the table, one containing the amount of my bet, and one ...
3
votes
0answers
116 views

Relation between maximizer's derivative and maximizing function

Let $u(x)$ be a continous bounded function such that $u'(x) >0$, $u''(x) < 0$. Define $A(x) = -\frac{u''(x)}{u'(x)}>0$. Let $Y$ be a random variable with $\mathbb{E}|Y|<\infty$. I solve a ...
0
votes
0answers
41 views

How to go about optimizing this function? (Maximizing)

If we are given a fixed integer $N > 0$ of choices we can pick out of a pool of $k$ values $c_0, \cdots, c_k$ (with repetitions allowed and $c_i > 0 \forall i$) and we want to maximize the ...
3
votes
1answer
93 views

Existence of a general-purpose (almost) universal optimization strategy

From Wikipedia about interpretations of no free lunch theorem A conventional, but not entirely accurate, interpretation of the NFL results is that "a general-purpose universal optimization ...
2
votes
1answer
302 views

Understanding no free lunch theorem

From Wikipedia: $Y^X$ is the set of all objective functions $f$:$X$→$Y$, where $X$ is a finite solution space and $Y$ is a finite poset. The set of all permutations of $X$ is $J$. A random ...
0
votes
1answer
115 views

Suitable Loss function for Order preserving Factoring of a matrix?

(Old-Question) Given a $n\times n$ symmetric matrix $X$, I would like to factor it using a vector $c$ of size $n \times 1$ such that: $\sum_{i,j} [X_{ij} \cdot c_i\cdot c_j]$ is minimum. How can I ...
3
votes
2answers
1k views

One vs multiple servers - problem

Consider the following problem: We have a simple queueing system with $\lambda%$ - probabilistic intensity of queries per some predefined time interval. Now, we can arrange the system as a single ...
1
vote
1answer
97 views

What is the maximum value of the minimum number of balls per bin?

$S$ people, $N$ bins, each person has a given subset of bins he can cover, each person is given $t$ balls. Question: What is the maximum value of the minimum number of balls per bin? i.e., allocate ...
2
votes
1answer
100 views

An optimal regression problem/proof

I want to find a function $f$ that given $x$ will predict $y$. The expected prediction error of $f$ is $$e = E[(Y-f(X))^2]=\int \int [y-f(x)]^2 p(x,y) dx dy$$ the expectation of $(Y-f(X))^2$ with ...
10
votes
3answers
1k views

Best fit ellipsoid

Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is ...
4
votes
3answers
586 views

Optimally combining samples to estimate averages

Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement ...