2
votes
2answers
57 views

Optimize rate of collection in counters

Suppose you have $K$ counters. The value of these $K$ counters are all $0$. Every second, each counter has a $J$ chance of incrementing itself, up to a max value of $I$. Every second, you may choose ...
1
vote
2answers
77 views

Maximizing discrete probability

I'm stuck with the following problem: Let's assume we have two buckets: bucket one contains $k$ white spheres and $l$ red spheres. Bucket two contains $n-k$ white spheres and $n-l$ red spheres (n a ...
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 ...
1
vote
1answer
45 views

Binomial Coefficient: monotonically decreasing in this range?

relating to this question, I'd like to ask a further one. Again we have $$f(x)={k-1 \choose x-1} p^x (1-p)^{k-x}$$ We know that this term is maximal for $x=kp$, before increasing, afterwards ...
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 $x$ ...
0
votes
1answer
27 views

Maximum payoff for safe bet

I'm having a hard time choosing a good strategy for this problem: assume that you have $m$ money that you can bet on $n$ mutually exclusive outcomes, all with unknown probabilities, and that each ...
1
vote
1answer
17 views

Maximum likelihood estimator transformed parameter

I don't get the gist of b). What is it that we are in fact calculating here? I don't get why we can just plug in the rearranged formula.
1
vote
0answers
43 views

Maximize the expected values of a function with constrain

Consider $p_1,p_2,...p_N$ are probabilities arranged in ascending order. $n_1, n_2,...n_N$ are numbers which are arranged in geometric progression. I want to Maximize E= $\sum\limits_{i=1}^N p_i\cdot ...
1
vote
1answer
41 views

need help with zero sum game

Tom chooses an integer in {1,2,3} and Bob chooses an integer in {2,3,4}. If the chosen numbers are the same, no money changes hands If the numbers are different the person who picks the bigger number ...
0
votes
1answer
7 views

Decomposition of chance constraint optimization problem

I want to decompose a chance constraint optimization problem and the constraint is: $Pr\left( \sum_{i}^{}{\left( x_{i}+\xi _{i} \right)}\leq c \right)\geq 1-\epsilon $ where $\xi _{i}$ are ...
1
vote
1answer
39 views

Minimizing the risk of misfires and duds in a missile control system

I was thinking the other day about all the different ways humanity could end itself -- I won't depress you all by listing them here -- and misfired nuclear missiles came to mind. The problem below is ...
0
votes
0answers
22 views

Constructing Matrix with Normal Distribution

I have a vector given whereby each element of the vector is assumed to be the average of one of a matrix' rows. Now I want to construct the matrix belonging to this vector, whereby the elements of the ...
0
votes
0answers
34 views

Estimation of random walk maximum and minimum positions

I am trying to prove that, if a simple and symmetric random walk $S$ starts at $S_0 = 0$ and finishes at $S_n = N$ with $N > 0$, then if there is a maximum $M > N$ and a minimum $B < 0$ (both ...
2
votes
2answers
36 views

Maximizing Expectation

A bag contains $b$ balls in total, $r$ of which are red, while the rest are white. In a game a player removes balls one at a time from the bag (without replacement). He may remove as many balls as he ...
1
vote
0answers
15 views

Characterizing limit of value functions in a stochastic control problem

Consider a probability space $(\Omega, \mathcal F , \mathbb P)$, $(B_t)_{t\geq0}$ M-dimentional brownian motion adapted to a filtration $(\mathcal F_t)_{t\geq0}$ over $\Omega$. In this context ...
1
vote
0answers
19 views

Determining the optimally scoring move on a probabilistically represented 2D grid in real time

I'm posting this to StackOverflow, cstheory.stackexchange.com, and math.stackexchange.com because I'm not really sure where it fits best. I hope that's OK. I have a 2D grid (size varies per map, ...
0
votes
0answers
239 views

Gradient-descent and Hidden Markov Models

I would like to use gradient-descent to fit the parameters of a simple 2-state HMM. This paper Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983), An Introduction to the Application of the ...
1
vote
0answers
85 views

Super Bowl Math Problem

A lot of us buy Super Bowl boxes this time of year. Here is my question. Suppose that we have the usual 10 by 10 grid of Super Bowl boxes where the coordinates labels for the horizontal and vertical ...
0
votes
0answers
41 views

Analytical computation of one Gaussian mixture model from another

I'm wondering if there is a way to analytically compute the optimal GMM (for a specific number of gaussians) in the case of approximating another GMM. E.g., is there an optimal single gaussian that ...
0
votes
0answers
35 views

Why is Expectation Maximization algorithm guaranteed to converge to minimum, even local?

I have read a couple of explanations of EM algorithm (e.g. from Bishop's Pattern Recognition and Machine Learning and from Roger and Gerolami First Course on Machine Learning). The derivation of EM is ...
1
vote
1answer
97 views

Optimal Strategy for Chosing Lottery Tickets

You have 2 types of lottery tickets: one that costs $c_1$ and has a probability of winning of $p_1$, and the other costs $c_2$ and has a probability of winning of $p_2$. The goal, as you might expect, ...
2
votes
1answer
570 views

Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
2
votes
0answers
65 views

Optimal elevator placement

I was thinking about this in my building today. Assume that the number of people trying to go up an elevator in a certain time period (say, an hour) is given by a Poisson distribution with mean $A$, ...
0
votes
0answers
55 views

GA (Genetic Algorithm) and stochastic simulation to solve optimization in R

My problem is to solve the following optimisation problem using GA (Genetic Algorithm)and stochastic simulation. The goal is to solve the maximisation problem : \begin{equation*} \begin{aligned} ...
0
votes
0answers
33 views

Probability of Sampling Matching of Two Independent Schemes in Time

I have encountered a very specific problem, that I don't even know the keywords to search for it. Here is the explanation: There are two independent sampling schemes like in the image below: I want ...
3
votes
1answer
71 views

Optimal consumption policy

I start with an initial capital C and at the beginning of day $n=1,...,N$ I observe the random variable $X_n$, where $\mathbb E X_n=\mu_n$. The $X_n$ are independent. I also choose $c_n$ on day $n$, ...
0
votes
0answers
7 views

LQ regulation scalar deterministic

Take a scalar deterministic linear system $x_t = Ax_{t-1} + Bu_{t-1}$ with cost function $\sum\limits_{t=0}^{h-1} Qu_t^2 + x_h^2$. I need to show from first principles that in terms of time to go $s$, ...
0
votes
0answers
40 views

Dynamic programming backward recursion

A stock broker can impress his boss if immediately after a week when the S&P moves either up or down he correctly predicts this is the last week in the calendar year that it moves in that ...
4
votes
1answer
119 views

Maximising probability for financial advice

I have the following problem: A financial advisor tries to impress his clients if immediately following a week in which the ftse index moves by more than $5\%$ in some direction he correctly ...
0
votes
0answers
41 views

Maximization of The Likelihood Function of Vector Entries and Its Norm

I'd be happy for assistance with the maximization of the likelihood function of the following model. The Parameters Vector $ \mathbf{\Theta} = [{x}_{1}, {x}_{2}] $. The measurement vector is $ ...
1
vote
2answers
50 views

Thief, exponential reward, optimal strategy

A thief robs a house every night. His profit each night is independent of others, and is a random variable with $Exp(1/\lambda)$ distribution. Every night, there is a probability $0<q<1$ that he ...
0
votes
1answer
35 views

Chance $U_1$is bigger than all other random variables

Please, help with the following would be highly appreciated. Again, I have an idea and a solution, but would like to see what other people think. Let $X_1, X_2,\dots, X_N$ be iid random variables ...
2
votes
1answer
148 views

Minimizing a specific function over n variables

Experimenting with something related to probability theory I came across the following $n$-variable function $$ f(p_1,\ldots,p_n) = \sum_{i=2}^{n-1} \left ( (1-p_i)(1-p_{i-1})^i (1-p_{i+1})^{n-i} + ...
1
vote
1answer
52 views

Guessing Game Stochastic Optimization

This is part of another post I did, but I think it has interest in its own right: Let $Y =\{X_{1},X_{2}...X_{N}\}$ be a set of $N$ random quantities with assocated set of distributions ...
1
vote
1answer
42 views

Maximize probability of working with the smartest person in a group of people.

My biggest regret in choosing courses in university was choosing statistics over probability. Hence, I have a problem approaching this question, and fear my skills in probability are insufficient. ...
1
vote
0answers
61 views

Can a condition for a global maximum (of some specific function) be given?

Suppose we have a twice continuously differentiable function $h(x) := \frac{g(x)}{1 - \delta + \delta F(x)}$, $0<\delta<1$, defined on the interval $[0, a]$ (where $a$ may be infinite). The ...
0
votes
0answers
81 views

A stochastic programming with a chance constraint

Let $X$ be a bounded positive variable with an unknown probability density function (PDF) and $f(X)$ be a differentiable positive function. $$\begin{align*} &\min/\max ...
1
vote
1answer
105 views

Gradient ascent, log likelihood

Good day, hi, would like to ask a question. If you have some spare time please kindly enlighten me on the following question. Gradient ascent: $=\sigma \leftarrow \sigma + \dfrac{d}{d\sigma} p(y\mid ...
3
votes
1answer
69 views

Exchanging $\min$ and expected value

Consider the inequality $\min \limits_{x} E_{\xi} \{Q(x,\xi)\} \ge E_{\xi} \{\min \limits_{x} Q(x,\xi)\},$ where $x$ is a vector of binaries, $\xi$ is a discrete random variable, and $Q(x,\xi)\ge ...
4
votes
1answer
265 views

Maximum of absolute value of linear combinations with i.i.d random variables

Suppose $x_{1},\dots,x_{n}$ are i.i.d random variables with density $p(x_{i})=exp(-|x_{i}|)/2$. Denote column vector $x=(x_{1},\dots,x_{n})^{T}$ Let $C\in\mathbb{R}^{n\times n}$ be a matrix with unit ...
1
vote
1answer
118 views

maximize the expected value of the logarithm of the weighted average of random variables

I'm trying to do the following. $$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$ where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
1
vote
0answers
45 views

Conditions for the ground state of Gibbs ensemble not to be “degenerate”

I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions. I am looking for books or other resources ...
0
votes
1answer
51 views

Optimization of entropy for fixed distance to uniform

Suppose that I know that a probability distribution with $n$ outcomes is very close to being uniform (that is: $\forall i,p_i=\frac{1}{n}$), and in particular for $n\epsilon\ll 1$ the distribution ...
0
votes
1answer
41 views

Optimize winnings in a money making game.

So, given a continuous random variable A (with some density and CDF function), and a value I choose V, what is the equation to determine the best value V to maximize my earnings given that I will be ...
1
vote
3answers
944 views

gradient descent optimal step size

Suppose a differentiable, convex function $F(x)$ exists. Then $b = a - \gamma\bigtriangledown F(a)$ imples that $F(b) <= F(a)$ given $\gamma$ is chosen properly. The goal is to find the optimal ...
2
votes
1answer
125 views

Optimal strategy puzzle

Play a game with an urn. $75$ blue balls. $25$ red balls. $1$ yellow ball. you get a dollar for every red and if you select the yellow you lose everything. what should be your strategy in the game. ...
1
vote
0answers
128 views

Facets of the convex hull as solution of an optimization problem?

Given $N$ points $x_1, x_2, ..., x_N \in \mathbb{R}^n$, consider their convex hull $$\mathcal{C} = \text{conv}( \{ x_1, ..., x_n \} ) = \bigcap_{j=1}^{J} \{ x \in \mathbb{R}^n : \ A_j x \leq b_j \} ...
0
votes
1answer
41 views

What is the optimal stopping point for an experiment when expecting unknown event

Assume we notice that stock prices are rising and we can deduce we are in a bubble. Assume we start at $w(0)=0$ worth at time $t=0$ and the value grows linearly with time $(w(t)=t)$. We know that ...
0
votes
2answers
77 views

Anyone saw this interesting function before?

Say $\theta\in\Re^n$ and $\theta_i\in(0,1)$ for all $i$. Define $$ f(\theta) = \frac{1}{n}\sum_i^n\{(1-\theta_i)\log(1-\theta_i)+\theta_i\log\theta_i\} $$ It is easy to see the minimizer of ...
2
votes
3answers
128 views

Packing radios into cartons - why is my solution wrong?

A manufacturer of car radios ships them to retailers in cartons of $n$ radios. The profit per radio is $\$59.50$, minus shipping cost of $\$25$ per carton, so the profit is $59.5n-25$ dollars per ...