Questions tagged [probability]

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

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Meaning of "Overdispersion" in Statistics

I am trying to understand what "overdispersion" means in statistics. Based on the Wikipedia page, "overdispersion" is defined as follows : "In statistics, overdispersion is ...
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If $\theta_1,...,\theta_k \sim D_k(a_1,...,a_k)$ so $\theta_i,\theta_j \sim D_2(a_i,a_j)$? [closed]

I'd like to know if $\theta_1,...,\theta_k \sim D_k(a_1,...,a_k)$ so $\theta_i,\theta_j \sim D_2(a_i,a_j)$ where $D_{(.)}$ is any multivariate distribution, for example, $D_k$ as k-variate normal ...
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How do I calculate the adjusted chance level for a classifier, if the features also affect the chance level?

If I have 5 different classes, and split the data into 70% training and 30% testing making sure each class is represented equally in both. Then I extract features from them in a One Vs. All approach, ...
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Why the part inside the integral is normal with mean t and sigma 1?

Why the part inside the integral is normal with mean t and sigma 1 because Y is standard normal so why (y-t)^2 becomes N(t,1).
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1answer
53 views

Number of tries until Failure with n different independent Bernoulli experiment

I have 3(n) coffee machines in an office. I have a historical log of these machines, and I know in the last ten days(t) how many times they failed to make a coffee. For example: machine 1: 10 ...
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13 views

Find expected number of events happening given that events are dependent

So, my question frames like this: There are n houses in row and there is a street light in between the houses. So, n-1 street lights. The glowing of street light depends on both the immediate houses ...
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1answer
44 views

Extreme Value Theory and Floods

Let's start with an example problem case. Say we measure two variables that are non-normally distributed and correlated. For example, we look at various rivers and for every river we look at the ...
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2answers
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What are the properties of the distribution of medians from samples of a set of integers?

Given a set of integers $S = \{1, 2, 3, ..., n\}$ I draw uniformly at random without repetition $m$ integers a large number of times ($1 \leq m \leq n$). For each of these samples (each of size $m$) I ...
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28 views

Joint densities do not exist, compute probability

I am taking a course in probability and in one of the PS I have got the following description: $Y_i \sim \exp(\lambda_i)$ for $i=1,...,n+1$. Define $X_i = \min(Y_i,Y_{n+1})$. In the first subsection ...
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Max number of vote per person [closed]

Population = 75 4 cnadidatets for election Each voter may choose to cast 1 2 3 or 4 votes. Max 1 voter per candidate. What is the max possible # & % of vote per candidate ?
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1answer
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What is meant by relative frequency in calibration curves

After reading docs on scikit learn on the probability calibration there's couple concerns that bug me. I don’t really understand how the curve values are calculated (y-axis) namely the frequencies. ...
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Conventions for pmf/pdf

The way I usually see the conventions: for a continuous random variable $X$, the pdf is denoted $f_{X}(x)$ or $f(x)$,where $x$ is an instance of $X$. And for a discrete random variable $X$, the pmf ...
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Simulation in R to check by histograms that the marginal distribution are correct? [closed]

The distribution on $R^2$ with joint density h with respect to the Lebesgue measure is: $$h(x,y)=\frac{3}{2}y 1_{A}(x,y), \ \ A=\{(x,y) \in R^2|0<y, x^2+y^2<1\}$$ I have found that marginal ...
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1answer
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How can we allocate when we have 150 open slots every day (5 days a week) for those 200 arrivals every day

My question is to solve a very basic problem related to the allocation of slots. Say there are 20 teams with 10 persons in each team. I have 150 open slots every day (5 days a week) for those 20 teams ...
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3answers
158 views

Geometric conditional Probability

In order to start a game, each player takes turns throwing a fair six-sided dice until a $6$ is obtained. Let $X$ be the number of turns a player takes to start the game. Given that $X=3$, find the ...
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1answer
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Three Envelopes Problem

In a game we have 3 envelopes, A, B, C, with a random number (from 1 to 100) sealed inside each. Nobody knows what number is inside. We win the game if we find the highest number, using only one of 2 ...
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Variational coin tossing from scratch: calculation of the expected log likelihood

I'm working my way through this tutorial about variational inference for a coin tossing. Let's say the probability of the event head is denoted by ...
3
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1answer
323 views

Regression Models when the Covariates have many Zeros

While researching this topic, I have come across different regression models which allow for the response variable to have many zeros. This includes: Negative Binomial Regression Zero Inflated ...
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27 views

What determines y-axis scaling on a normal probability plot?

I've been investigating Q-Q plots, specifically the normal probability plot, to determine if a set of data I have is approximately normal. I.e., a plot of the data should be approximately a straight ...
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1answer
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Absorption Time of Continuous Markov Chains

I have the following question about the Absorption Times of Markov Chains in Continuous State-Space. I was reading the following article on Absorption Times of Markov Chains (https://en.wikipedia.org/...
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2answers
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What is the difference between a realisation of a random variable and random variable itself?

I am having a hard time distinguishing random variables from their realisations. (Please note that for the sake of simplicity of my question I use discrete values in my example below.) Usually, a ...
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"Automatically" detect biased subsets in probability distribution

Background: Suppose we have a model generating probabilities conditional on a state vector - for simplicity we can just assume the outcome is 0 or 1 (imagine for example simple logistic regression): $...
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Understanding Covariance after Variance (visually)

2 Points i understood from variance derivation- A) For calculating Variance we do not subtract (or mod add), but rather sum squared all points' differences from the mean. B) Variance of 1,2,3 will be ...
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Is it possible for a set of random variables to each be highly correlated with another variable, but not highly correlated with each other? [duplicate]

Let $X_1, ..., X_n$ and $Y$ be random variables. Is it possible for the $X_i$'s to all have a high magnitude of correlation (absolute value of Pearson's $r$) but not be strongly correlated with each ...
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The correct interpretation of null hypothesis significance testing NHST when comparing two means. Concise, pragmatic, and correct definitions [closed]

Please review the following statistical scenario, which is then followed by an attempted statistical interpretation. Please review the interpretation for accuracy, and if it is in any way incorrect, ...
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Joint Probability in R [closed]

I have a finite sequence of discrete values X = {x1, x2, x3...xn} and I want to compute the joint probability of each pair of elements in the sequence. i.e, P(x1, x2), P(x2, x3) etc. How should I ...
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1answer
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Expectance of predictive posterior on binomial-beta modeling

I'm struggle with figuring out how to prove: $E(\tilde{x}|x)=n\frac{a1}{a1+b1}$ where $x|\theta \sim Bin(n,\theta)$, $\theta \sim Beta(\alpha,\beta)$ and so $\theta|x\sim Beta(a_1=\sum{x_i}+\alpha,a_2=...
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1answer
57 views

MGF of the product of a exponential and a bernoulli random variable

Let $𝑍=π‘‹π‘Œ$ , where $X$ and $Y$ are independent, 𝑋 ~𝐸π‘₯π‘π‘œπ‘›π‘’π‘›π‘‘π‘–π‘Žπ‘™(0.01) and π‘ŒβˆΌπ΅π‘’π‘Ÿπ‘›π‘œπ‘’π‘™π‘™π‘–(0.3) Is there a way to find the m.g.f of 𝑍? I know that I can find the C.D.F by doing as ...
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2answers
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Binning Calibrated probability scores for business use

Context: We have a model that outputs calibrated probability scores for a binary classification problem (events/nonevents). There is a general business requirement that we bin these outputs further to ...
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24 views

Connection between the exponential distribution and extreme value distribution

I found here a justification that the connects the exponential distribution and standard extreme value distribution that goes as follows. Let $X\sim \text{Exp}(\lambda)$ and $Y=\log(X)$. Now $\lambda(...
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1answer
38 views

The product of 2 independent markov chains

Let $(X_n)_{nβ‰₯0}$ and $(Y_n)_{nβ‰₯0}$ be two independent Discrete-time Markov chains $\in Z$. a) Define the process $Z_n = (X_n, Y_n)$ $\in Z^2$. Is $(Z_n)_{nβ‰₯0}$ a Markov chain? b) Define the process $...
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34 views

Statistical Models that "Exploit" Distributional Knowledge of the Predictor Variables

I am trying to see if there any Statistical Models that (better) "Exploit" Distributional Knowledge of the Predictor Variables. For example, I feel that is a common misconception (e.g Where ...
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0answers
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The Elephant in The Room: How is Real-World Domain Knowledge Converted into Bayesian Priors?

I have been trying to look into the daunting problem within Bayesian Models: How is Real-World Domain Knowledge Converted into Bayesian Priors? Logically speaking, it seems that Bayesian Priors can be ...
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1answer
31 views

What would be the proper distribution to model the number of particles in a state in canonical ensemble

Suppose my system has $N$ particles, and I want to find a distribution for $n_i$, the number of particles in the $\epsilon_i$ energy state. What I do know is the boltzmann probability, which tells me ...
3
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1answer
71 views

How is it possible to have $P(A|B \cup C)$ lower than both $P(A|B)$ and $P(A|C)$?

Let's say we investigate disease probability given two symptoms. Thus we have 3 variables: A) has disease B) has symptom 1 C) has symptom 2 We have following data: Now we want to see which symptom ...
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1answer
17 views

Why Scipy does not have geom.fit but has norm.fit?

This is a newbie question. I see that Scipy can estimate the two norms (mean and variance) of a normal distribution using sample data. That's done using the ...
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1answer
32 views

Probability calculus on random bitstrings

If I randomly choose 2 bitstring on length (n) with n to be an even number, what is the probability, parametrized on n, that at least n/2 of the bits are equals? In my mind, since the random choice of ...
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68 views

Manually Deriving the Maximum Likelihood Estimates for Less Common Probability Distributions

I have a question relating to Manually Deriving the Maximum Likelihood Estimates for Less Common Probability Distributions. Suppose we generate 200 random numbers from a Normal Distribution ~ (1,2): <...
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1answer
15 views

Continuous state markov chain [closed]

Given a discrete time - continuous state Markov chain Y, how do I estimate the transition probability P(y t+1 | yt) ? Which estimation technique should I use for such a series? Could someone help me?
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10 views

Statistical Inference on Covariates (Instead of the Response Variable)

In statistical modelling, it seems as though we are always more interested in predicting the expected value of the response variable conditional on some observed vector of covariates. However, are ...
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0answers
10 views

Graph Neural Network and Protein Structure Prediction [closed]

Does it fit that kind of issues? Say, if I describe resulting protein as a graph since a source amino acid chain has some transition probabilities.
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1answer
25 views

Generative Models - Class Conditional Density and Posterior Probability

In the section 1.5.4 of Bishop's PRML book, a brief description of generative models is given. For classification decisions, it is stated that "the class- conditional densities may contain a lot ...
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The Realizability Assumption [closed]

I'm reading the the first chapter of Understanding machine learning from theory to algorithms: I do not understand the meaning of the following assumption: definition 2.1 (The Realizability Assumption)...
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0answers
16 views

Estimating conditional entropy of a series

I have a time series $Y$ (continuous values) and I want to estimate its conditional entropy such that $H(y_{t+1}|y_t)$ represents the conditional entropy of $Y$ at time $t+1$ given its value at time $...
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1answer
37 views

Independence of Cause and Mechanism - Causality

In causal modelling, we say that $A \longrightarrow B$ if forcing a value change A will influence the likelihood of $B$ while holding all other variables in the system constant. We call this a direct ...
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23 views

How to calculate exponential growth when the value is a percentage or probability?

I have a variable x that takes values between 1 and 0 (0<x<1). I want to calculate value of ...
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0answers
17 views

Expectation of inverse of sum of i.i.d. positive variables [duplicate]

Description: There is an indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time ...
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1answer
33 views

Do Probability Distributions Inherently "Capture More Information" Compared to Hyperplanes?

Do Probability Distributions Inherently "Capture More Information" Compared to Hyperplanes? A regression model (whether linear or polynomial) is said to be represented as hyperplane through ...
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1answer
34 views

Summarizing Non-Normal Probability Distributions

Suppose I have the following data: ...
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1answer
24 views

Probabilistic interpretation of sum of quantile functions

We know that the weighted sum of CDF $$ F(x) = w_1 F_1(x) + w_2 F_2(x), \,\, w_1 + w_2 = 1 $$ is the CDF of the mixture distribution. Is there a probabilistic interpretation for weighted sum of ...

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