Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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Determine the expected value and the probability of some events (applicative probability problem)

Suppose that the time that a phone call lasts is a random variable with density function given by: $$ f(t) = \begin{cases} \frac{1}{5} e^{-\frac{1}{5}t} & \text{if } t > 0 \\ 0 & \...
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Why is $P(A \mid C \cap B) = P(A \mid C)$ true in this instance?

As I was reading through this paper http://www.jstor.org/stable/25652278 I came across the following problem: Consider an urn with $N$ colored balls, the number of red balls, $X$, has a binomial ...
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Given any events $A$,$B$ and $C$ where $A$ and $B$ are independent, is it true that $P(A|B\cap C) = P(A|C)$?

It seems to make intuitive sense that if the events $A$ and $B$ are independent then $P(A|B\cap C)=P(A|C)$ because the occurrence of event $B$ should not change the probability of event $A$ even when ...
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Probability problem - Jackpots

The rules of the final round of King's Lottery in Byteland are simple: The player starts to draw balls from an opaque bag containing $a$ red balls, $b$ green balls and $c$ blue balls initially. After ...
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What probability would I use in a negative binomial distribution to say the expected number of false identifications per true identification?

I have binary classifications made by some model, and I know the truth. From this, I can calculate the confusion matrix and all associated values: true positive rate, true negative rate, false ...
Dave's user avatar
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How to approach problem [closed]

I have a discrete random variable f(phi) with known probabilities for each n outcomes. Phi represents the random variables m parameters. Each parameter is its own independent discrete random variable ...
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6 votes
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What is the probability a difference of two appears before 3?

Given a sequence of trials, where each trial is rolling two dice, each trial's outcome is the absolute difference between the values on the die. What is the probability that the outcome of 2 occurs ...
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If random variables X,Y are independent is $P(X>k)*P(Y>z)=P(X>k,Y>z)$? [duplicate]

If random variables X,Y are independent is $P(X>k)*P(Y>z)=P(X>k,Y>z)$? I know if X and Y are independent then $P(X=k)*P(Y=z)=P(X=k,Y=z)$
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Do I need to calibrate a model to use the Brier score?

When I use the Brier score loss, do I need to calibrate the model and then use the calibrated model's predictions as input into the Brier score loss? If I just use a non-calibrated model's ...
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How does Dempster-Shafer Theory of Evidence relate to Deep Learning?

I am reading this article and it has the following phrase - "Dempster-Shafer Theory of Evidence assigns belief masses a set of classes (unlike assigning a probability to a single class)". ...
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Interview Question: What is the probability they will be home in more than 30 minutes?

The following is an interview question: A student leaves Univeristy (U) to walk Home (H). It is a distance of 4 blocks in a straight line. At each crossing, they toss a coin deciding whether to move ...
Ria's user avatar
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Statistics of lottery

I have a dozen donuts and 84 friends. Since there is not enough to go around and no one wants to share, I will put all 84 names in a hat (a hat that perfectly randomizes) and draw 12 names. What are ...
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Gambler Ruin Probability

I read the document about sequential testing written by Evan Miller on the website (https://www.evanmiller.org/sequential-ab-testing.html). In his document, he mentioned that the probability of the ...
Grace Xu's user avatar
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Why is it complex to write Non-Normal Distributions in Multivariate form? [duplicate]

In statistics, it's pretty straightforward to write a formula for the joint probability distribution of two random Normal Variables: \begin{align*} x_1 &\sim \mathcal{N}(\mu_1, \sigma_1^2) \\ x_2 &...
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Find function $f$ such that the performance of models $h_1$ and $h_2$ is similar

Suppose that $h_1$ and $h_2$ are the first and third order polynomials respectively, which are obtained by solving the OLS equation (using the training dataset). Also consider the following ...
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Order agnostic autoregressive sequence probability

Autoregressive models define the following probability of a sequence $x$ as $p(x)=\Pi p(x_t | x_{<t})$ using the probability chain rule. Then Hoogeboom et al. 2022 state that if we uniformly sample ...
JustBlaze's user avatar
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How to understand the definition of Markov Chain $P(X_{n+1}\in B\mid \mathcal{F}_n)=p(X_n,B)$?

The definition of Markov Chain in Durrett (Probability: Theory and Examples, 2019, Section 5.2) is: $$P(X_{n+1}\in B\mid \mathcal{F}_n)=p(X_n,B), $$ where $p$ is the Markov transition kernel ...
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KL Divergence, which pdf is which?

Let $P(x)$ and $Q(x)$ be two pdfs. Let us say that $P(x)$ is the original baseline distribution and $Q(x)$ is the model (or estimate) distribution. I wanted to take the KL Divergence (or the 'distance'...
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Why is the multivariate normal distribution is $(\Sigma, C)$ sub-gaussian?

The definition of sub-gaussian from a book I work with is: $X\in\mathbb{R}^n$ is $(\Sigma,C)$ sub-gaussian if $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\in\...
Torben I's user avatar
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Goodness-of-fit test for very skewed data [closed]

I am working with two quite large datasets (9000 obs and 3800 obs). Each observation has been grouped into 1 of 12 categories and I am looking to test if the frequency of categories in the smaller ...
Elin Sjoeholm's user avatar
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Likelihood ratio as minimal sufficient statistics in infinite parameter space

I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first: Consider a family of density functions $f(x|\theta)$ where the ...
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Convergence in product of sequence of random variables

This question has been addressed previously in here but my question is different. It is from Hogg and McKean's "Introduction to Mathematical Statistics". Theorem $5.2.7.$ Let $\{X_n\}$ be a ...
TryingHardToBecomeAGoodPrSlvr's user avatar
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Group aggregates of conditional probabilities

I suspect I don't know the academic or SEO word for what I'm looking for. Put simply, I want to know a group's total expected success rate given each member's likelihood of attempting the "trial&...
Brandan's user avatar
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drawing from multiple urns based on a random number

Suppose I have $N$ numbered urns, each with different colored balls. Now I have some fixed discrete distribution that I use to draw a random number $i \in \{1 ... N\}$. I now pick a ball from urn $i$, ...
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Random walk on cube with condition of not returning

An ant is placed in a corner of a cube and cannot move. A spider starts from the opposite corner, and can move along the cube's edges in any direction (x,y,z) with equal probability 1/3. On average, ...
Charlie's user avatar
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Likelihood of sum of log-normal and normal distribution [closed]

Given $y(x_t)=e^{f(x_t)}-\varepsilon _t$ with $\varepsilon _t\sim N(0,4e^{f(x_t)})$ and $f\sim GP(\mu,\sum)$. What is the likelihood $p(y|f)$? Is it $p(y|f)\sim N(e^{f(x_t)},4e^{f(x_t)})$? Thanks a ...
manhtr76's user avatar
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Bayes's theorem problem issues

Problem Statement: Assume that 40% of all interstate highway accidents involve excessive speed by at least one of the drivers (event $E$) and that 30% involve alcohol use by at least one driver (event ...
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Question about mean hitting times

A homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with discrete state space $\mathcal{S}$. Set $$\tau_{k}:=\text{inf} \left\{n\ge 0:\, X_n=k \right\}.$$ where $\tau_{k}$ is defined to be $+\infty$,...
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Expectation propagation: $g_i(\theta)\sim\mathcal{N}(\mu_i,\Sigma_i)$ but $\Sigma_i^{-1}$ is non-invertible

I paste here the section on Expectation Propagation from Gelman et al.'s Bayesian Data Analysis (pg 340). I have two questions: In Step 5 (with the red box), where we infer the hyperparameters of the ...
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When transforming t in the joint probability density f (x, t), i.e. t=1/(mt+1), how does the marginal probability density f (x) transform?

for example, if: $$\begin{aligned} f_{X_{pi},T_{pi}}(x,t|*)=& \begin{aligned}\frac{\pi s^2}{\alpha_{pi}^2}\exp\left(\frac{\alpha_{pi}(x-\frac12)\nu_{pi}}{s^2}-\frac{\nu_{pi}^2}{2s^2}(t-\tau_p)\...
shenwanggong's user avatar
2 votes
2 answers
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Difference between Probability and Effects in Logit Models

when I run a logistic model I get log odds that I can easily convert to probabilities. What I don't understand is how can I use percentages instead? Here's the code: ...
Luca's user avatar
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Distributing n candies to n children

What is the probability of any child getting two or more candies? My thinking is that if there is one child getting 2 candies, that means there is one child not getting any candies. So $(n-1/n)^n$ ...
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Problem with Cumulative distribution function

I can't understand this cumulative distribution function. I would like to calculate the data distribution function: ...
Zollikofen4's user avatar
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1 answer
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A question related to the convergence of mathematical expectations restricted to an Interval centered on zero

Let $(X_j)_{j= \mathbb 0}^\infty$ a fixed realization of strictly stationary AR(1) process: $$X_j = 0.9 \,X_{j-1}+ \eta_{j}, \quad (\eta_j) \overset{iid}{\sim} N(0,1)$$ For each $n$, consider $B_n\sim ...
André Goulart's user avatar
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Percentile calculation for binary attributes

Assume I have the following dataset: ...
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4 votes
1 answer
387 views

Where does Quasi-Likelihood formula come from?

In regular likelihood/log likelihood, if there is random variable "$Y$" with pdf (probability distribution functions) $f_Y(y)$... the likelihood of this can be written as: $\mathcal{L}(y_i) =...
Uk rain troll's user avatar
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The relationship between Bellman Equation and EM(Expectation Maximisation) Algorithm

Today I came across Value Evaluation Algorithm under the topic of Bellman Equation when learning reinforcement learning, which is stated as: While not converged: Policy Update: $\pi_{k+1} =\arg\max_{...
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5 votes
3 answers
484 views

Show that $\frac{1}{n(n-1)} \sum_{i\neq j} \sin\left(X_i X_j\right)$ converges almost surely to a constant

Let $X_i$ be iid random variables. How does one show that $$ \frac{1}{n(n-1)}\sum_{i\neq j}^n \sin\left(X_i X_j\right) $$ converges almost surely to a constant?
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Understanding three prisoners problem [closed]

I briefly state the problem: There are 3 prisoners. One of them is going to be executed. Prisoner A asks the warden to tell him name of 1 prisoner who will not be executed tomorrow. Warden tells him ...
Nima's user avatar
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3 votes
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Lower bound on probability that pairwise independent Bernoulli random variables sum to 1

I'm trying to find a lower bound on the probability that $k$ pairwise independent Bernoulli random variables with $p=\frac{1}{k}$ sum to 1. The probability that they sum to $>1$ is upper bounded by ...
Murey Tasroc's user avatar
2 votes
1 answer
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Proof Check -- Convergence of degenerate random variables equivalent to convergence of sequence

This is from Hogg and McKean's "Introduction to Mathematical Statistics". At the stage this exercise (number $5.1.1$) is given, he has not yet spoken about convergence in distribution etc. ...
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Applying filter (e.g., moving average) on Binomial distributed random values

I start with Binomial distributed random values with known $N$ and success probability $p$. I can easily estimate PMF and CDF of such distribution. Now assume these Binomial distributed random values ...
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3 votes
2 answers
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What is the probability of being dealt two pairs when all $\binom{52}{5}$ poker hands are equally likely?

Question from Sheldon Ross's First Course in Probability, Chapter 2: If it is assumed that all $\binom{52}{5}$ poker hands are equally likely, what is the probability of being dealt two pairs? (This ...
Roy's user avatar
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How to graphically show how "normal" a collection of dice rolls are in a game

I was trying to think of a way to show how "normal" a set of dice rolls are in a game like catan. In Catan, each turn you roll 2d6 and the sum of those 2d6 dictate game outcomes. A game can ...
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Algorithm to find the probability of a number given the probability of another number?

Let me preface that I'm not sure if this is even possible and I'm unsure of what math principals would apply but I'm trying to build an algorithm for a sports betting project that can find the ...
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Qustion about transient states (Continuation)

Let $$f_{kk}^{(n)}:=\Pr(X_n=k,X_v\ne k,1\le v\le n-1\mid X_0=k),~n\in \mathbb Z^+ .$$ Attributing to the comments of @Zhanxiong , I have added the other two cases to the Case 1. Case 1. Is there a ...
Kevin's user avatar
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how to assess how likely it is that a candidate gets more than 50% of the total votes [closed]

So far in a country are counting votes and 72.9% of the votes have been counted. Candidate A has 43.8% of the votes, candidate B has 43.54%. Assuming that there are not a significant difference ...
Mario Alberto Jaimes's user avatar
0 votes
1 answer
82 views

Best rigorous book on probability theory for ESL [closed]

I know this is similar to many questions asked here before, but I couldn’t find question that addressed what I need. I started ESL but I’m having a lot of trouble with the probability theory part of ...
1 vote
1 answer
66 views

A question about the definition a p-values [duplicate]

In hypothesis testing, the definition of p value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is ...
AJP's user avatar
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Calculating expected trials before "X out of the previous Y" trials reach a threshold?

Sorry; not a statistician; I don't know how to correctly word this. This came up with X (formerly Twitter)'s Community Notes system. If at any time 3 out of 5 of your previous community notes are ...
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