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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

19,111 questions with no upvoted or accepted answers
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41
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977 views

Limit associated with a recursion

If $z_n < 2y_n$ then $y_{n+1} = 4y_n - 2z_n$ $z_{n+1} = 2z_n + 3$ Else $y_{n+1} = 4y_n$ $z_{n+1} = 2 z_n - 1$ Consider the following limit: $$\lim_{n\rightarrow\infty} \frac{1}{n}\left(z_{n+1}...
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240 views

Brownian motion and Beta distribution

I am interested in the distribution of the time that the standard Brownian $W_t$ motion on $[0,1]$ satisfies the following inequality: $$W_t \ge stW(1)$$ For different values of $s$. I conjecture that ...
14
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482 views

Random sum in coupon collection

I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
13
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594 views

Progressive Dice Game

$(2019.)$ Edit: Rewriting the question to make it clear. The progressive dice game At the start, you have a fair, regular six sided dice $D=(1,2,3,4,5,6)$. The game is played for $n$ turns. ...
12
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365 views

Expected number of operations on a vector until one of the coordinates becomes zero.

Let's say we have a vector $v = (x_1, ..., x_n) \in \mathbb{N}^n$ where $x_1 = x_2 = ... = x_n$. Next we choose an ordered pair of coordinates at random $(i, j)$ where $i, j \in \{1, ..., n\}$ and $i \...
12
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494 views

What is the Probability of Transmission Between Two Nodes in a Neural Network?

I have a network which is an Erdős–Rényi graph. It is a simple neural network with degree 0.7N where N is the number of nodes. Each weight between neurons is 1/N, meaning that if node n has fired ...
11
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692 views

Calculating growth rate of a population of Minecraft chickens

I have a rather strange question (for this Stack Exchange anyway). It felt too mathematical to ask elsewhere. If this is out of place here, please let me know. A chicken in Minecraft lays eggs; ...
11
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0answers
577 views

At what rate does the entropy of shuffled cards converge?

Consider a somewhat primitive method of shuffling a stack of $n$ cards: In every step, take the top card and insert it at a uniformly randomly selected one of the $n$ possible positions above, between ...
11
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0answers
845 views

How many edges does an Erdős-Rényi graph have to have, to almost surely have a component with multiple cycles?

An Erdős-Rényi graph is a random graph, selected according to the distribution obtained one where we have some number $n$ of nodes, and some probability $p$ of each potential edge being present....
11
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464 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
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179 views

Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $af(G) = \frac{|\{(g, a) \in G \times Aut(G)| a(g) = g\}|}{|G||Aut(G)|}$. Equivalently it can be defined as $P(A(X) = X)$, where $A$...
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386 views

Extracting an (almost) independent large subset from a pairwise independent set of Bernoulli variables

Let $n>1$, and let $X_1,X_2, \ldots ,X_n$ be non-constant random variables with values in $\lbrace 0,1 \rbrace$. Let us say that a subset of variables $X_{i_1},X_{i_2}, \ldots,X_{i_d}$ is complete ...
9
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281 views

Number of simultaneously solvable linear equations with nonnegative variables

Let $N$ variables $x_i \ge 0$ but not all of them be zero. One may fix $\sum_{i=1}^N x_i = 1$. There are $P$ equations which need to be solved, with coefficients $a^k_i$ indexed with superscripts $k =...
9
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149 views

Generating function for number of $r$-disjoint subsets each of size $k$

Fix $n, k$. Then $$ C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!} $$ is the number of ways to form $r$ disjoint subsets each of ...
9
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272 views

How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
9
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128 views

Asymptotic value of card drawing game

A deck consisting of $r_0$ red cards and $b_0$ black cards is randomly shuffled. The host turns up the cards one at a time; if it is red, you get $\$1$; otherwise you pay the host $\$1$ (and you're ...
9
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1answer
197 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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2k views

Random graph connectivity, and the existence of isolated vertices

Here $G_{n,p}$ represents the Erdős-Rényi random graph model, where the graph has order $n$ and each edge is added independently with probability $p$. I am faced with proving the following claim: ...
9
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421 views

Does this calculation have a name, or a generic formulation?

Background Informatiom I would appreciate help in identifying or explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: Sample from the distribution of each of $i$ parameters, ...
8
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0answers
253 views

Average number of strings with edit distance at most 4

Consider a binary string of length $n \geq 4$. An edit operation is a single bit insert, delete or substitution. The edit distance between two strings is the minimum number of edit operations ...
8
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0answers
179 views

Is it true that $\phi(\mu)=\mu F(\mu)^2-\int_{\mu}^{\overline{v}}F(v)[1-F(v)]dv\geq 0$?

Consider a random variable $V$ with distribution function $F$ and density function $f$ with support $[\underline{v},\overline{v}]$, where $0\leq\underline{v}<\overline{v}$. The mean is $\mu$. Here $...
8
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0answers
260 views

Find $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$

Now also asked on MathOverflow. Let there be $n$ pairs of shoes in a box. The the probability that from the $r \le n$ shoes I am taking out of the box there are exactly $p$ pairs is given by \begin{...
8
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0answers
385 views

Implications of inequalities

For $i=1,2,3$, consider a random variable $Y_i$ taking value in $$ \mathcal{Y}:=\{(1,1), (1,0), (0,1), (0,0)\} $$ and a random closed set $S_i$ taking value in $\mathcal{S}$ that is the power set of $...
8
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0answers
209 views

What is the probability that the Golden State Warriors will break the NBA regular season record of wins?

There are $82$ games in a regular season, and the current record is held by the Chicago Bulls, at 72-10. As of yesterday (March 4th 2016), the GSW season performance stood at 55-5. Assuming they ...
8
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0answers
2k views

Constructing an $\epsilon$-net of $l_2$ unit ball

I am interested in probabilistic or explicit ways to construct an $\epsilon$-net of the $l_2$ unit ball in $\mathbb{R}^{d}$. I know that, for every $\epsilon > 0$, there exists an $\epsilon$-net $\...
8
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0answers
172 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: \...
8
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0answers
1k views

Probability that at least one of four hands missing at least one suit

Deal each of four players a 13-card hand at random. What is the probability that at least one of the four hands is missing at least one suit? Let $A_i$ mean that player $i$ is missing at least one ...
8
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0answers
450 views

Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
8
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0answers
1k views

Is kernel density estimation a GMM with uniform mixture weight?

recall that for a Gaussian Mixture Model, the density of p(x) (multivariate) is $$P(x) = \Sigma_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ On the other hand, non-parametric density estimation ...
8
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698 views

How do Kolmogorov 0-1 law and CLT imply normalized sample mean doesn't converge in probability nor a.s.?

From WIkipedia the central limit theorem states that the sums Sn scaled by the factor $1/\sqrt{n}$ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-...
8
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1answer
380 views

Using Jensen's inequality to prove another inequality?

Suppose $u(\cdot)$ and $v(\cdot)$ are two differentiable, strictly increasing, and strictly concave real functions. Specifically, $v(\cdot)$ is "more concave" than $u(\cdot)$ in the sense that there ...
7
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0answers
75 views

Bound on Expectation and Variance of Random variables ratio with positive weights

Let $X_1, X_2, \dots, X_n$ be $n$ strictly positive iid random variables with bounded variance. Let $w_1, w_2, \dots, w_n$ be non-negative deterministic constants such that $\sum_{i=1}^{n} w_i = 1$. ...
7
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0answers
97 views

How to distribute $n$ red balls between two bins to maximize the chance of sampling only red balls?

Consider two bins that contain an unknown number of black balls. We wish to split $n$ red balls (i.e., choose a number $x\in\{0,\ldots,n-1\}$ to place in a random bin) so that if we pick one ball at ...
7
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0answers
93 views

The probability of rolling $N$ 10-sided dice and forming groups that add at least 10

I'm trying to answer this question for an RPG game. The player (or the GM) has to roll $N$ 10-sided dice and then she has to form groups that sum at least 10 (I'll call them Raises, using the game ...
7
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0answers
147 views

Events A and B are independent such that $P(A)=6P(B)$

Events A and B are independent such that $P(A)=6P(B)$ and $P(A \cup B) =0.915.$ Find P(B). I know that $P(A \cup B)= P(A) + P(B) -P(A)P(B)$ Then $0.915=6P(B)+P(B)-6P(B)P(B) \\ \rightarrow 0.915=7P(...
7
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0answers
152 views

Flies in a cube

Two flies sit in the corners $(1, 1, 1)$ and $(n, n, n)$ of a discrete cube $M^3$, where $M=\{1,\ldots, n\}$. After every unit of time both flies decide randomly and independently of each other ...
7
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0answers
413 views

Interpreting the Lindeberg's condition

I know the Lindeberg's CLT but I don't have a good grasp of the intuition behind the Lindeberg's condition. Could you please give some intuition behind said condition via an example (or, perhaps, via ...
7
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0answers
180 views

Supposing joint normality, is a pair of asymptotically uncorrelated sequences also asymptotically independent?

Let's say there are two sequences of random variables $(X_n, Y_n)$ and we know that For each $n$, $(X_n, Y_n)$ is normally distributed. $\mathrm{cov}(X_n, Y_n) \rightarrow 0$ as $n \rightarrow \infty$...
7
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1answer
201 views

Density given by variable-coefficient PDE

I am looking for a time-dependent probability density $f(x,y,t)$ solving the equation $$-\frac{\partial f}{\partial t} = \alpha\cdot \big(y - F(x)\big)\frac{\partial f}{\partial x}+\beta\cdot \big(G(y)...
7
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2answers
210 views

Probability of an Indisputable winner at Texas Holdem

What are the fraction of hands that can be classified as "indisputable winners" (aka "the nuts") after the river is revealed in Texas Holdem? By "hand" I mean the 2 hole cards you have that no one ...
7
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0answers
635 views

Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random Variable)...
7
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1answer
1k views

Two-sided hitting time of Brownian motion

I am trying to compute the hitting time of a linear Brownian motion on a two-sided boundary. More specifically, let $W_t$ be a (one-dimensional) Wiener process. Let $T = \inf \{t: |W_t| = a \}$ for ...
7
votes
1answer
197 views

Probability no two pairs are grouped together twice in a row?

There is a room with 48 people divided into 16 groups of 3 in round 1. In round 2, the group is again randomly divided into 16 groups. What is the probability that no two groupmates in round one are ...
7
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0answers
113 views

Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let $$ h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}| $$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = \...
7
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1answer
3k views

Fingerprint match probability

I am trying to use the formula for the birthday paradox as a reference to figure out an equation that represents the probability of a fingerprint match. Here's the equation for probability of a ...
7
votes
1answer
789 views

Probability that a random edge coloring of the complete graph is proper

Suppose we color the edges $\{1,\ldots, {n \choose 2}\}$ of the complete graph on $n$ vertices with $m$ colors each edge being assigned a color picked uniformly at random from $\{1,\ldots, m\}.$ I ...
7
votes
1answer
222 views

How can I find the limiting value of a time-dependent PDE?

I've managed to reduce a question in probability to the following simple looking PDE: $$ u_t = -t u_x + \frac{1}{2} u_{xx}, {\rm ~for~} x>0, \, t \in \mathbb{R} \;, $$ with a limiting initial ...
7
votes
1answer
240 views

If $Y$ is a nonnegative absolutely continuous random variable and $E[X|Y]=Y/2$, is $E[X|Y=-1]=-1/2$? Is $E[X|Y=2]=1$?

One of the definitions I learned for $E[X|Y=y]$ is the following: $$ E[X|Y=y]=\int_{\mathbb{R}} x\,P_{X|Y=y}(dx), $$ where $P_{X|Y=y}$ a probability verifying $$ P(X\in A, Y\in B)=\int_B P_{X|Y=y}(A)\...
7
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1answer
254 views

Recurrence of a certain class of $2$-$d$ random walks

As is well known, a symmetric random walk on $\mathbb{Z}^d$ (the lattice of $d$ dimensional vectors with integer components) is recurrent if and only if $d=1,2$. In particular it is transient for $d=3$...
7
votes
1answer
227 views

How to calculate probability of these two pair-of-sums ($S_{n}$ and $T_{n}$) and ($SEvenF_{n}$ and $SOddF_{n}$) being the same?

Say we have a sequence of $n$ positive integers, we can assume they're randomly chosen, let's call it $U_{n}$. Let $S_{n}$ = sum of $U_{n}$ from $1$ to $n$. Let $T_{n}$ = sum of $n$ from $1$ to $n$. ...