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Questions tagged [probability]

For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].

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Probability mass function for combinatorics problem

The question is as follows: A small pond contains 15 fish: 10 blue fish and 5 red fish. A set of 5 fish are caught at random. (Once a fish is caught it is not placed back in the pond.) Let K be the ...
Jacob's user avatar
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Finding References for known results about mixing time of a Markov chain arising from Gaussian elimination

I am looking for references for the known results on the following problem. Let $(X_t)$ be a Markov chain on $GL_n(F_2)$, where in each step, an ordered pair $(i,j)$ is chosen uniformly at random, and ...
mathematico's user avatar
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Stochastic Gaussian Process vs Non-Stochastic Gaussian Process?

Lately I have observed that there are two types of Gaussian Processes: Type 1: https://www.cs.toronto.edu/~rgrosse/courses/csc411_f18/slides/lec20-slides.pdf Type 2: https://en.wikipedia.org/wiki/...
konofoso's user avatar
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-1 votes
1 answer
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Probabilities I'm gonna win a tournament when I'm better at one game and bad at another

I'm playing a game of chess and go. It's a three round game and to win I have to win two rounds in a row. I'm better at chess and my opponent is better at go. I can chose to play go-chess-go or chess-...
whyu's user avatar
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0 answers
7 views

Gaussian Distribution vs Gaussian Noise?

I am trying to understand Donsker's Theorem (https://en.wikipedia.org/wiki/Donsker%27s_theorem): Let $F_n$ be the \textit{empirical distribution function} of the sequence of i.i.d. random variables $...
konofoso's user avatar
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19 views

Marginal distribution is normal, but the joint distribution is not. Why?

Denote the CDF of standard normal distribution as $\phi(\cdot)$. The joint CDF of $(X,Y)$ is $ F_{X,Y}(x,y) = \phi(x)\phi(y)[1-\alpha(1-\phi(x))(1-\phi(y))]$ where $\alpha \in (-1,0)\cup(0,1)$. ...
Kaven Lin's user avatar
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1 answer
17 views

Method of Moment for Normal mixtures $p\cdot N(0, 1) + q\cdot N(\eta, 1)$

Setup Let $X_1,\ldots , X_n$ be random variables according to $$ p\cdot N(0, 1) + q\cdot N(\eta, 1),\ p\in (0, 1), q:=1-p. $$ We use method of moments to obtain the needed starting $\sqrt{n}$-...
ytnb's user avatar
  • 562
2 votes
1 answer
31 views

$\mathbb E(\max(X_1,...,X_{t+1})|\mathcal{F}_t)$ where the $X_i$ are iid uniform

Let $X_1,...,X_T$ be independent and identically distributed uniform random variables on $[0,1]$. Let $$M_t:=\max\{X_1,...,X_t\},$$ $L_t=M_t-ct$ for a $c>0$ and $L_0:=-\infty$. If $\mathcal{F}_t=\...
Analysis's user avatar
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I'm quite unsure how to approach part (d). Would greatly appreciate any advice. Thanks in advance!

I'm unable to embed. Here's the link to the question
John Maclay's user avatar
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FOB Poker: Probability of a sequence of numbers (with the existence of "wild-card" number)

At work we have 2 factor authentication using a fob that generates a sequence of 6 numbers 0-9. We started playing "poker" by having everyone generate a number at the same time and see who ...
tender_bits's user avatar
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25 views

Why are logarithms used to measure Similarity?

I was reading this link on how to test the difference between 2 probability distributions :https://chjackson.github.io/kl.html. If $P(x)$ and $Q(x)$ are discrete distributions, the KL Divergence can ...
wulasa's user avatar
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What is the meaning of $p_{g_i}$ in this equation?

I am reading "The element of statistical learning" and having some question regarding equation 2.36. The book stated that: "A more interesting example is the multinomial likelihood for ...
alksdhalksjdb's user avatar
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Find a majorant of two random variables

Let $X,Y \in \mathbb{L}^{\infty}(\Omega,\mathcal{F}, \mathbb{P})$ two random variables such that $\mathbb{P}(X<0)>0$ and $\mathbb{P}(Y<0)>0$. Is it true that there exists a $Z \in \mathbb{...
leobgg's user avatar
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I dont get how i can solve b [closed]

I tried solving it with the zentraler Grenzwertsatz and Normalverteilung but i cant get so be near the solution. Can you please help me so i understand it enter image description here enter image ...
it seems's user avatar
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Asymptotic distribution involving a high-dimensional matrix

Let $ \varepsilon \sim \mathcal{N}({\bf 0}, I_n)$ be a n-dimensional multivariate normal RV and $x_1,x_2,...,x_n \sim \mathcal{N}({\bf 0}, I_p)$ be i.i.d p-dimensional multivariate normal RVs. $X$ is ...
V. V's user avatar
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0 answers
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Balanced cricket simulation game engine

I have a cricket sim game that I'm building and I'm looking for help to create a more balanced game. I suspect there are some common ways this could be approached, so looking for any feedback on the ...
bwash70's user avatar
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0 answers
70 views

Pi's digits (unlikely occurrences?) [closed]

There are no zeros in the first 31 digits of pi. There are many unlikely patternes in pi like repeating numbers, are these coincidences?
Zellion's user avatar
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-1 votes
1 answer
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Probability space of a random sample and almost sure convergence

When working with random samples in statistics, there are 2 approaches: To have the single-outcome space $(\Omega, F,P)$, on which $n$ i.i.d. random variables $(X_1,...,X_n)$ act. I.e., a single $\...
jose89's user avatar
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5 votes
1 answer
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Evaluating $ \lim_{n\to \infty} \frac{1}{n}\int_0^n \max(\{x\},\{\sqrt{2}x\},\{\sqrt{3}x\})dx $, where $\{x\}$ is the fractional part of $x$

I found an interesting problem on the internet: $\lim_{n\to \infty} \frac{1}{n}\int_0^n \max(\{x\},\{\sqrt{2}x\},\{\sqrt{3}x\})dx$, where $\{x\}$ is the fractional part of $x$. I have known that this ...
Zhou_Key_Error's user avatar
1 vote
1 answer
28 views

Counting squares in a regular polygon with $4n$ vertices ($n \geq 2$)

Select at random $4$ vertices from a regular polygon with $4n$ vertices ($n \geq 2$). Find $n$ providing that the probability that the $4$ chosen vertices are the vertices of a rectangle that is not a ...
ten_to_tenth's user avatar
3 votes
2 answers
87 views

Want an example of uncorrelated but dependent joint Bernoulli example [closed]

Can anyone give an example (joint probability table) that two Bernoulli variables are uncorrelated but not independent?
roz's user avatar
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0 answers
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What does it mean that the mean squared error of a test data is smaller than the training?

I was working with a set of data. I used bayesian regression to derive a model for it and I get the following results. MSE for bayesian regression testing data : $0.14611199672315536$ MSE for bayesian ...
User's user avatar
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10 votes
1 answer
41 views

Understanding Nash Equilibria in a Bimatrix Game

I am currently studying game theory and I came across a problem involving a bimatrix game. The bimatrix is given by: $$ (A, B) = \begin{pmatrix} (4, 2) & (0, 0) \\ (0, 0) & (1, 3) \end{...
鈴木悠真's user avatar
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36 views

Number of ways to draw $3$ cards from $26$ cards, numbered $1-26$, so that the difference between any of the two cards drawn is at least $2$

A box contains $26$ cards, numbered from $1$ to $26$. Draw $3$ cards at random from the box. How many ways to do this so that any two of the cards drawn have numbers whose difference is at least $2$? ...
ten_to_tenth's user avatar
1 vote
1 answer
36 views

Help me calculate the triple summation

Problem We consider $$ \sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\xi_i)(x_{j\nu}-\xi_j) = \sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\bar{x}_i)(x_{j\nu}-\bar{x}_j) - 2n \...
ytnb's user avatar
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0 votes
1 answer
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Are unconditional probability and conditional probability method necessary to generate the same result?

mcs.pdf shows 2 methods to calculate the probability of "switch wins" for Monty Hall problem. unconditional probability (See Figure 17.5): So we have the probability $\frac{1}{9}\cdot 6=\...
An5Drama's user avatar
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1 vote
0 answers
30 views

Stratonovich multiplication

I’m currently studying a paper Brownian heat engine with active reservoirs, and I have encountered some problems regarding Stratonovich multiplication. This is a part of the content in the article: $$...
Joker's user avatar
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0 votes
1 answer
52 views

CLT for an insurance company

An insurance company has 1,000 individuals of the same age insured. The probability of death in a given year for each insured individual is 0.01. The insured pay an annual premium of 1,200 eur, and in ...
marek's user avatar
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0 votes
1 answer
33 views

Prove symmetry of probabilities given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution as $$F(y) :...
BCLC's user avatar
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0 votes
1 answer
37 views

Distributing candies problem: Seeking verification of my solution

There are 20 identical candies. How many ways to divide the candies among 3 children so that: Every child has at least one candy Every child has at least 2 candies The first child has at least 1 ...
ten_to_tenth's user avatar
0 votes
2 answers
28 views

Finding the distribution of the median of three independent random variables

Problem: Let $Y_1$, $Y_2$ and $Y_3$ be independent and uniformly distributed over the interval $(0,1)$. Let $Y_0$ be the median of the three variables. Find the probability density for $Y_0$. Answer: ...
Bob's user avatar
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2 votes
0 answers
35 views

Flipping a "Dynamic Coin" until the First Heads

Here is a coin flipping game I thought of: Initialize: Set lambda=5, p=0.5, set=1, and generate num_flips from Poisson(lambda). For each set: Do the following until heads is observed: For each flip ...
konofoso's user avatar
  • 453
1 vote
1 answer
38 views

May the sum of Wiener processes be a Wiener process?

May $X_t = W_t + \tilde{W}_t$ be a Wiener process, if $W_t, \tilde{W}_t$ are Wiener processes? I know that: $X_t$ may be not a Wiener process, e.g. in case $W_t = \tilde{W}_t$. If $W_t$ and $\tilde{...
Sergei Nikolaev's user avatar
0 votes
1 answer
58 views

If $X\sim N(\mu, \sigma^2)$ and $\Phi$ is the CDF of a standard Normal random variable, what is the distribution of $\Phi(X)$?

Let $\Phi$ be the cumulative distribution function of a standard Normal random variable, $Z\sim N(0,1)$. Let $X\sim N(\mu, \sigma^2)$ follow any Normal distribution. We know that $\Phi(Z)\sim \...
cgmil's user avatar
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0 votes
0 answers
18 views

Item selection probablity after sampling m items out of k and then sampling two item from the m items uniformly.

$m$ items are drawn from $[k] = {1,2,3,...., k}$ according to probability $q(i)$ iid with replacement. That is the probability of item $i$ being drawn is $q(i)$. Then two items $x$ and $y$ are drawn ...
Mario420's user avatar
0 votes
1 answer
18 views

Does the deterministic sequence dominated by a random sequence converges to zero in deterministic sense if the random sequence converges a.s.

I am new to the concepts of different types of convergence of random sequences. Suppose $\{a_k\}_{k\in\mathbb{N}}$ is a deterministic sequence. Let $\{X_k(\omega)\}_{k\in\mathbb{N}}$ be a random ...
curiosity's user avatar
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1 vote
0 answers
40 views

How to prove that $X_n \stackrel{\text{a. s.}}{\rightarrow} 0$

Let $\left\{X_n\right\}_{n \in \mathbb{N}}$ be a strictly decreasing sequence of positive random variables such that $X_n \stackrel{P}{\rightarrow} 0$. I have to prove that $X_n \stackrel{\text{a. s.}}...
Cyclotomic Manolo's user avatar
1 vote
2 answers
91 views

What can I say about $P(B|A_1 \cap A_2)$ if I know that $P(B|A_1), P(B|A_2) \approx 1$?

Let $(\Omega,\Sigma,P)$ be a probability space and $A_1,A_2,B \in \Sigma$. Define $P(B|A_1) = E(\chi_{B}|A_1)$. From real life, I have experience that if there are two numbers $\alpha_1,\alpha_2 \...
Mikke Mus's user avatar
  • 149
1 vote
0 answers
24 views
+100

Smallest eigenvalue of non-central Wishart matrix

Suppose that $X \in \mathbb R^{d \times n}$ is a random matrix with independent entries, each of which follows the standard normal law $\mathcal N(0, 1)$, and that $M \in \mathbb R^{d \times n}$ is a ...
Roberto Rastapopoulos's user avatar
-5 votes
0 answers
31 views

Can someone explain the concept used in this video? [duplicate]

https://youtube.com/shorts/ao_u8Oxthlc?si=4OsKPk_eO4vJihW4 How are the chances 66.67% later because now there are only 2 doors and one with the car. So shouldn't it be 50% chance to win?
NOTE Book's user avatar
2 votes
1 answer
48 views

Probability that each male sits opposite to a female in two opposite 4-seat rows

$4$ males and $4$ females are randomly directed to $8$ seats, arranged into $2$ opposite rows (each row has $4$ seats). What is the probability that each male sits opposite to a female? The key is $\...
ten_to_tenth's user avatar
0 votes
0 answers
38 views

Why is the probability of drawing exactly one bad product in a sequence of three draws the same, regardless of which draw is the bad product?

A box contains $10$ good products and $5$ bad products. Draw at once $3$ random products from the box. What is the probability of drawing exactly $2$ good products? Certainly, the answer is $\dfrac{\...
ten_to_tenth's user avatar
1 vote
0 answers
24 views

Expected Time for Bus to Arrive based on Changing Probabilities?

Suppose there is a bus arrival with the following properties: At schedule = 1, the bus is expected to arrive according to some random number generated from an exponential distribution with $\lambda$ =...
konofoso's user avatar
  • 453
-1 votes
0 answers
16 views

Why can we get the equation $p(x) = E[p(x+Y)] + o(h)?$ [closed]

$X(t) = B(t) + \mu t$ is a Brownian motion process with drift coefficient µ. We now compute some quantities of interest for this process. We start with the probability that the process will hit $A$ ...
Garin's user avatar
  • 1
1 vote
1 answer
29 views

How to bound the total number of trials for a Binomial distribution if number of success is given?

Suppose I am sampling a random variable that follows a binomial distribution $Binomial(n, p)$. The sample rate $p$ is known. Now, after the sampling, I got $m$ success cases. How do I find the upper ...
picker's user avatar
  • 53
1 vote
0 answers
45 views

Prove or disprove: $P(A\cap B|C)=P(A\cap B)$ given $P(A|C)=P(A)$ and $P(B|C)=P(B)$. [duplicate]

Prove or disprove: $P(A\cap B|C)=P(A\cap B)$ given $P(A|C)=P(A)$ and $P(B|C)=P(B)$. Since \begin{align} P(A \cap B|C)=P(B|A \cap C) P(A \cap C)/P(C)=P(B|A \cap C) P(A), \end{align} the statement is ...
kenji's user avatar
  • 33
1 vote
1 answer
26 views

Writing the Lagrange of a Normal Distribution

I am watching this Youtube video on how to derive the Multivariate Normal Distribution using the Principle of Maximum Entropy (https://www.youtube.com/watch?v=7qsB8ElrCC4 @ 3:13). Here, the Lagrange ...
konofoso's user avatar
  • 453
0 votes
0 answers
18 views

Existence of probability distributions whose averages have sizeable deviations from the mean?

This question is a follow-up question to this old question of mine and this partial solution. Question. Does there exist a (real or even integer) random variable $X$ with finite first moment $\mu$ ...
Olivier Bégassat's user avatar
-1 votes
0 answers
21 views

The probability of randomly generating finite abelian groups

While reading this article (https://msp.org/involve/2013/6-4/involve-v6-n4-p04-s.pdf), I had a question: why in the last expression, where $$G = \mathbb{Z}_{p^{n_{1}}}\oplus\mathbb{Z}_{p^{n_{2}}}\...
moonruleni9ne's user avatar
0 votes
1 answer
50 views

How do I calculate p-value in this problem?

Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below: Population 1: $63, 72, 72, ...
No Go's user avatar
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