Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. 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.

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41
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690 views

probability for a $n\times n$ matrix to have no complex eigenvalues

Let $A$ be a $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has no complex eigenvalues? The answer cannot be 0 or 1, since ...
15
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255 views

Probability of a group being finite

Suppose $F_m := F[x_1, … , x_m]$ is a free group on $m$ generators $x_1, … , x_m$ and lets define Cayley ball $B_m^n := \{e, x_1, x_1^{-1}, … , x_m, x_m^{-1}\}^n$ as the set of all elements with ...
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746 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 ...
14
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397 views

The good, the bad and the ugly with conditional probability/expectation

I thought that I understand conditional probability and expectation until I saw this question: The problem for conditional expectation. Basically, it is given that: $$(X,Y)\sim f(x,y)=\begin{cases} ...
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875 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. ...
14
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1k 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....
13
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234 views

Poisson distribution with an integer $\lambda$ value

I have noticed that when a Poisson distribution has an integer value of $\lambda$, the following holds: $$ \mathbb{P}[X = \lambda] = \mathbb{P}[X = \lambda - 1] $$ I have been able to prove this ...
13
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0answers
637 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 ...
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599 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 $\frac{1}{N}$, meaning that if node n ...
12
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0answers
415 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 \...
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348 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 ...
11
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2k 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 ...
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498 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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192 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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329 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 and answered affirmatively there. 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 ...
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3k views

Rigorous Proof of Slutsky's Theorem

I was hoping to type up my proof of Slutsky's Theorem and get confirmation on the excruciating details being all correct... Statement of Slutsky's Theorem: $$\text{Let }X_n, \ X,\ Y_n,\ Y,\text{ share ...
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2answers
241 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 ...
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1answer
280 views

Randomly Generate Probability Mass Function With Specific Entropy

How can I randomly generate a probability mass function such that the entropy of a random variable that follows that probability mass function is a specific value $h$? Basically, I need to randomly ...
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0answers
900 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)...
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408 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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234 views

$|𝐸[𝑋]|+𝜌(𝑋)≥1$?

Suppose $X_1, X_2, ... \sim X$ are i.i.d. random variables on $\mathbb{Z}$. Then the sequence $\{P(\sum_{i=1}^{d(X)n} X_i = 0)^{\frac{1}{d(X)n}}\}_{n=1}^\infty$ converges to some constant $\rho(X) \in ...
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103 views

Two random variables $X,Y$ such that $X+Y$ has the same distribution as $X$

Let $X$ and $Y$ be two almost surely finite real-valued random variables which are not necessarily independent. Assume that $X$ is non-negative and $Y$ has a finite, positive mean. Is it possible that ...
9
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1answer
698 views

Tighter tail bounds for subgaussian random variables

Let $X$ be a random variable on $\mathbb{R}$ satisfying $\mathbb{E}\left[e^{tX}\right] \leq e^{t^2/2}$ for all $t \in \mathbb{R}$. What is the best explicit upper bound we can give on $\mathbb{P}[X \...
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168 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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0answers
186 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: \...
9
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1answer
419 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where $b,\...
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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 ...
9
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1answer
892 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-...
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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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445 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, $\...
9
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1answer
533 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 ...
8
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0answers
168 views

Probability that $\int_0^tX_s\,dW_s$ lies within $1/t$ of $X_t$

Consider the inequality $$f(x)-\frac1x\le f’(x)\le f(x)+\frac1x$$ on the positive axis. This tells us that $f(x)\sim e^x$ with infinitesimal deviation, and we can use identities such as Grönwall's ...
8
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0answers
188 views

$P(X_{(n-k_n)}>X_1\mid X_1>u_n)=0$?

Let $X_1, X_2,\dots$ be continuous random variables with full support (I need the result when they follow AR time series $X_i=\alpha X_{i-1}+\varepsilon_i$ for iid epsilons. But if you will consider ...
8
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0answers
171 views

Expected value of the maximum minimum set

Pick $m$ numbers in $[0,1]$, independently and uniformly at random. It is known that the expected value of the smallest number is $1/(m+1)$ and of the largest number is $m/(m+1)$. Now, partition the ...
8
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0answers
209 views

Proof of one-side version of Bennett-Bernstein inequality

I'm going to prove the following: For independent random variables $X_i$, $i \in [m]$ satisfying $X_i-E[X_i] \le b$ for some constant $b > 0$. Let $\bar{X} = \dfrac{1}{m}\sum_{i=1}^m X_i$, we have ...
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0answers
252 views

Is there any significance to this "doubly stochastic matrix" with both a discrete and continuous index?

This is just idle curiosity. Consider the function $(\lambda, n) \mapsto e^{-\lambda} \frac{\lambda^n}{n!}$, where $\lambda \in \mathbb{R}_{\ge 0}$ is a nonnegative real parameter and $n \in \mathbb{Z}...
8
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1answer
107 views

Betting game with numbers

Someone is challenged to play the following game: there are 36 marbles in an urn. 21 of them are red, 9 are blue and 6 are green. Each red is worth 0 rupees, each blue 100 rupees and each green 1000 ...
8
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0answers
241 views

Is this an okay way to calculate covid-19 death rates?

I'm a tad rusty on my math skills but this isn't too hard. Can you confirm that I am doing this right. I wanted to know what percent of people die from covid-19 so I started looking for statistics. ...
8
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0answers
231 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
votes
1answer
1k views

roll a dice repeatedly until the sum goes above 63

I saw this question online but I don't know if my solution is right or not. Here is the original question: Roll a die repeatedly. Say that you stop when the sum goes above 63. What is the probability ...
8
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0answers
107 views

Reversal of an Autoregressive Cauchy Markov Chain

Let $\mu_0 (dx)$ be the standard one-dimensional Cauchy distribution, i.e. \begin{align} \mu_0 (dx) = \frac{1}{\pi} \frac{1}{1+x^2} dx. \end{align} Suppose I fix $h \in [0, 1]$, and form a Markov ...
8
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0answers
305 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 =...
8
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0answers
169 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 ...
8
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0answers
678 views

Accounting for uncertainty in an Elo rating system for Foosball

For a Foosball game at work we implemented a rating system based on the Elo system. Allthough we achieved a sensible result so far, which provided us with a lot of fun (which is the goal) we feel we ...
8
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0answers
222 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
votes
1answer
251 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)...
8
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0answers
163 views

Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
8
votes
1answer
731 views

Why is the supremum a random variable in the Glivenko–Cantelli theorem

According to wikipedia: Assume that $X_1,X_2,\dots$ are independent and identically-distributed random variables in $\mathbb{R}$ with common cumulative distribution function $F(x)$. The empirical ...
8
votes
0answers
283 views

Expected area of an inscribed triangle in a sphere

On the surface of a unit sphere, three points $A$, $B$ and $C$ are chosen in the following way: Points $A$ and $B$ are chosen randomly and independently on the whole surface After $A$ and $...
8
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0answers
606 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...

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