Questions tagged [probability]

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

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Convergence to 0 in probability for non-iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k^2$, with $a_k > 0$ and $\sum_{k=1}^{\infty} a_k < \infty$. How can we prove that $S_p:= \frac{1}{p}\sum_{k=1}^{p}...
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What is intuition behind the product rule of probability and independent events?

I just bumped into a simple question. Let's say I want to compute the probability of taking both Math and Science courses (i.e., $P(M \cap S)$) given this information: Total class size is 10; 7 ...
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Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...
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Why are the $n^k$ ordered samples equally likely while the ${n+k-1}\choose{k}$ unordered samples are not?

Source: Blitzstein and Hwang’s Introduction to Probability. The following is said: “Consider a survey where a sample of size k is collected by choosing people from a population of size n, one at a ...
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How to draw a transition diagram for a time-homogenous Markov Chain [closed]

Any tips on how to calculate the probabilities when M and mods are involved:
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Deriving distribution for multiplayer game results from pairwise probabilities

Suppose there is a game with three participants: Player A, Player B, and Player C. One player will finish in first place, another in second place, and another in third place (no ties allowed). I know ...
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How to combine independent probabilities on parts of a series to get conditional probability over the entire series?

Given a series $X_{0-2}$ if I have independent probabilities at each step for the series having a particular value say [$p_0$, $p_1$, $p_2$], how can I calculate the total probability $P_{0-2}$ of the ...
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Can we always pull a joint posterior apart?

If we have a posterior distribution $p(A,B|\theta)$, is it always true that $p(A,B|\theta) = p(A|\theta)p(B|\theta)?$
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Queuing Arrangement, Probability

Waiting in a line for a Saturday morning movie show are 2n children. Tickets are priced at a quater each. Find the Probability that nobody will have to wait for change if before a ticket is sold to ...
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If $E[|X_n|] = O(n)$ is $E[|X_n|^2] = O(n^2)$?

Let $X_n$ be a random variable that depends on $n$ and suppose $E[|X_n|] = O(n)$. Then can we say $E[|X_n|^2] = O(n^2)$? If it doesn't hold in general, are there particular interesting cases where it ...
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Do Particle Filters actually approximate the posterior distribution?

Im reading a tutorial paper about particle filters (Link) in which it is stated that as the number of samples tends to infinity the approximated posterior density given by $p(x_k|z_{1:k}) \approx \...
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How to sample from different datasets such that they have similar distributions?

I have data from multiple datasets with the boxplot given below In the above figure, I have data from 7 different datasets. I am looking for a sampling strategy such that samples from each dataset ...
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Rewriting integral/summation as weighting estimator

I recently read a biostats paper which featured the following identity: $$ \sum_{y, l, m} y P(y, l, m \mid c, a) \frac{P(l \mid a, c) P\left(m \mid a^{*}, c\right)}{P(l, m \mid c, a)}=E\left(Y \frac{P\...
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Gaussian distribution with Kronecker product in the Covariance matrix

Assume we have two correlated multivariate Gaussian random variables $\mathbf{d_1}$ and $\mathbf{d_2}$ both distributed as $\mathcal{N}(\mathbf{0},\mathbf{R})$. We also know that $\mathbf{d_1}-\mathbf{...
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Unfair coin probability

This question has me looking like an idiot in my stats class and I need help this is the problem. I have an unfair coin that lands on tails 75% of the time, what is the probability that it dosnt land ...
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Bootstrap bias-corrected percentiles

Following [1] p. 185-186 the $BC_a$ (bootstrap corrected and accelerated) confidence intervals are given by: $$ (\hat{\theta}^{*(\alpha_1)},\hat{\theta}^{*(\alpha_2)}), $$ where $\hat{\theta}^{*(\...
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Law of the unconscious statistician, is there a condition on g?

What are the criteria that $g(X)$ needs to satistify before we can write $E(g(X))=\int_{x \in \Omega}g(x)f_X(x)dx$??
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Simple probability equations for a value appearing within a range

I have basic knowledge in probability theory, although, I'm unsure of the necessary equation and concept to use towards my current idea, and would really appreciate some experienced support towards ...
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Why is this situation describing independent events?

I am working through some practice questions in a statistics textbook and I am struggling to understand why I got this question wrong. This is the context for the question: The AAPOR is an ...
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What is the distribution of the time-to-ruin for a gambler's ruin problem that allows “pauper bets”?

In another question on this site I have derived the distribution for the time-to-ruin in the gambler's ruin problem where the wealth of the gambler follows a discrete-time random walk. In this ...
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Given a target # for the end of year, how can I calculate the probability of reaching the target given my pace at any given moment in time?

Let's say I have a target to reach 100M widgets by the end of the year. I need to monitor in real-time the probability of hitting that target. "What's the probability we'll make it today?" ...
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Applying Bayesian inference

I want to know if the application of Bayesian inference below is correct and if so, what is the next step. I'm considering buying an old house. Based on its age, I think it has a 75% chance of leaking ...
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Use of Change of Variables (in probability distributions) in Machine Learning

I am learning about machine learning from a probabilistic perspective via Kevin Murphy's so far fantastic Textbook (2021) Machine Learning - Probabilistic Machine Learning - An Introduction. I'm in ...
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Find the decision boundary in bernoulli distribution

I encountered this question but I find some obstacles to convert this problem into a t-test format, can you guys give me some hint? Thanks! Your local casino acquires some new slot machines, which the ...
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Simple Numeric example to understand Pointwise Mutual Information

I have been trying to find a simple numerical example for PMI in order to understand the calculations, but I have not been able to find one. I have the following data (assuming there are more data, ...
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How to prove the tail behavior of the sum of random variables with one dominating?

Assume I have given independent, continous random variables $X_1, \ldots, X_n$ and assume that they all have support $[-\infty, \infty]$. If $X_1$ asymptotically dominates all others, i.e. $$f_{X_i}(\...
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What are the cv.glmnet coefficients?

Let's say one performs a ridge logistic regression with cv.glmnet with standardized variables in R. Here, the outcome of the model is given in coefficients. ...
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Unsure of how $E_*\{\hat f^*(x;h) - f(x;h)\}^2$ is being explicitly evaluated in a paper on bootstrapping for density estimation

I'm reading a 1989 paper by Charles Taylor 'Bootstrap Choice of the Smoothing Parameter in Kernel Density Estimation' and I'm not sure how he arrived at a certain result. Let $X_1,X_2,\dots,X_n$ be a ...
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Most likely outcome under Bernoulli trials [closed]

If we have a Bernoulli trial with outcomes $A$ and $B$. Outcome $A$ occurs with probability $p$, and $B$ with probability $1-p$. I read that if $p<0.5$, (so for example, $p=0.499$), then the most ...
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Can someone Help me out to understand the lecturer says we're talking about Population quantiles, Population median wrt Probability density functions? [closed]

The definition of quantile given here is, Ath Quantile is at a point where the probability ( or area under distribution ) up till that point is A.
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What features can be extracted from a probability distribution? [closed]

I have been looking online regarding feature extraction and I am looking at extracting features from probability distribution by getting the characteristics of the distribution. I know that most ...
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Find the Probability density function of a mean random variable

lets say $Y_{1},\ldots,Y_{n}$ are simple random samples with the PDF: $f_{\theta}(y)=\theta y^{\theta - 1} \mathbb{I}(0 \le y \le 1) $ How can I find the PDF of $\bar{Y}$? is it even possible?
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Gaussian Distribution: How to calculate the Cumulative Distribution Formula (CDF) from the Probability Density Function (PDF)? + Error Function? [duplicate]

I understand that we can calculate the probability density function (PDF) by computing the derivative of the cumulative distribution formula (CDF), since the CDF is the antiderivative of the PDF. I ...
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The relationship between two probability mass function (poisson distribution)

There are two cylinder bottles with radius r1 and r2 was on the ground to collect rain drop.what is the relationship between the probability mass function of two bottle? I guess that each of the ...
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Is this notation , $\ell(y,f(x;\theta))= -\log p(y|f(x;\theta))$, correct for the negative log probability loss function of a classifier?

In Murphy’s Probabilistic Machine Learning: An Introduction, he states that the loss function for a probabilistic classifier $f(x;\theta)$ is the following: $$\ell(y,f(x;\theta))= -\log p(y \mid f(x;\...
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Bounds of the optimisation problem in NICE paper

In the paper https://arxiv.org/abs/1410.8516, a model called Non-linear independent component estimation is proposed. Following are the governing equations for the forward pass in the network. --------...
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CLT for non iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k$. I'm looking for CLT of the sum $S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$. Since $U_k$ are correlated, I'm looking ...
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Metrics for multiclass classification model accuracy

Usually the last layer in multiclass classification models is a softmax, which is essentially a vector with elements the confidences for each class. The standard top-1 accuracy takes account only if ...
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Estimators of location and scale versus mean and variance

This question is rather semantic than statistical. In Robust Statistics, estimators of mean and variance of a distribution are often called respectively "estimators of location" and "...
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joint probability and conditional probability

I came to ask this question because I couldn't understand an answer in What are the differences between "Marginal Probability Distribution" and "Conditional Probability Distribution&...
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Markov Chain for Amino Acid Sequence

I was wondering if it is possible to apply Markov Chain to this amino acid sequence "ddvlsldeddddsdyncgednd" to find the probability of this sequence forming in this particular positional ...
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Differences Between the Central Limit Theorem and Consistency

I have recently finished studying the central limit theorem and the idea of consistency. I am still a little fuzzy about them, so I was wondering what are some key similarities and differences of the ...
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Did I correctly apply the factorisation theorem in this example?

Suppose that we have a density $f(x,\theta)=c(\theta)\psi(x)\unicode{x1D7D9}_{x \in]\theta,\theta+1[}(x)$. Find a sufficient statistic for this density. My attempt : Since $\Bbb L(X,\theta)=c(\theta)^...
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How to model the probability of detecting an image, given it is seen multiple times

Are there any existing methods/models describing the probability of an object being detected by a computer vision algorithm given it is seen $n$ times at similar angles and orientations? I know that ...
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Probability of a Protein Sequence [closed]

I am trying to find a way to calculate how improbable for certain protein sequence to exist due to random chance. For example: Take this protein sequence: "ddvlsldeddddsdyncgednd" I want to ...
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Chances of testing negative

Buddy takes a test and tests positive for a disease. Buddy was close to $6$ other friends they all take the same test and end up testing negative. The test has a $FPR=0.01$ and a $FNR=0.15$. What's ...
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Markov model for patients on a transplant waitlist?

I am developing a Markov model in Treeage based upon survival data for individuals on a transplant waitlist (running first order monte carlo). Over the time horizon, individuals may either become ...
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Does lack of data affect covariance matrix estimate?

I am building some experiments using the multivariate normal probability density function to estimate the likelihood of a given sample to come from a distribution. For that, the PDF is built using as ...
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Calculate the sentiment knowing the single probabilities

I am working on finding the sentiment of some text. Right now I am using a python library called VADER that for each sentence it gives me back the probability that a specific phrase is positive, ...

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