# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $x$ ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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. ...
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 \} ...