Search Results

This is similar to Anthony Quas's answer but maybe even simpler. Let $X_n$ be a Binomial random variable with probability $p$ and $n$ trials. Then $P(X_n > k)$ is an increasing function of $n$. Wri …

One of the deepest and most beautiful coupling arguments I have seen is the proof by Moser and Tardos of their Algorithmic Local Lemma. This algorithm tries to find a good configuration of variables …

(Cross-post from math.stackexchange.com Q#166689) I would like to lower-bound $E[X Y]$ where $X, Y$ are two random variables such that: $X \in [x_0, 1], Y \in [y_0, 1]$ $E[X] = x, E[Y] = y$ $X \geq …

The paper "Randomness-Efficient Oblivious Sampling" by Mihir Bellare and John Rompel approaches this by applying a Chernoff bound. (See Appendix A)

I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has …

Let us suppose that we are in the setting of Janson's inequality for Poisson-type deviations of increasing events. Specifically, we have independent Bernoulli variables $X_1, \dots, X_n$, and events $ …

Consider a polynomial $p(x)$ with of degree $d$ with coefficients in $GF(2)$. How many roots can it have in $GF(2^m)$? The intent here is that $d \ll 2^m$. The trivial bound is of course $\leq d$. Wh …

In the course of a proof, I used the following principle, which seems so intuitive that it should have a name: Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) …

Thanks for all the interesting perspectives on this question. It seems that there are two quite distinct reasons for focusing on expected value. The first is based on the law of large numbers, and wor …

While it is not known that $P \neq NP$, it is known that for most oracles $A \subseteq \omega$ we have $P^A \neq NP^A$. This might be interpreted as "evidence" for the conjecture when $A = \emptyset$. …

I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression …

Given a graph $G$, I would like to determine a method for randomly generating subgraphs $G'$ with the following properties: Each edge of $G$ has at least some probability $p$ of going into $G'$ The …

(Cross-posted to math stackexchange question 130154) I am trying to analyze the following branching process. We start with a root (level 0) node. Each surviving node has two children, each of which m …

(This is a cross-post from Math StackExchange https://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities) Let $\vec X = (X_1, \dots, X_k)$ be a draw from a …

Suppose we have an Erdos random graph with $n$ vertices and $c n$ edges. What can you say about the probability that the graph is connected? (More importantly) If it is connected, what is the distr …