Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
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If a transformation preserves distance for ANY metric, must it be the identity?
I am trying to prove the following statement and I am looking for some guidance or a hint.
Let $X = \mathbb{R}^n$ and let $f: X \to X$ be an affine transformation defined by $f(x) = Ax + b$. We know ...
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Chi-squared random variable
Given a random variable $X$ which is $\chi^2_{n}$, can I define $n$ independent standard normal random variables $Z_{1,...,n}$ on the probability space such that $X = Z_1^2 + Z_2^2 + ... + Z_n^2$ ...
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Should tolerance bounds be used for inferring the probability that an event occurs?
I've encountered a statistical procedure that someone else did and I am asking for perspectives from others on whether this procedure is appropriate.
We have a measurement, $X$, and there is also a ...
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How can negative log likelihood be properly compared between two sets with different sample sizes?
I have a dataset that I have divided into training and testing data, with approximately 160 samples in the training set and 40 in the testing set. I fitted a probability distribution to each dataset ...
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Solving Poisson process fish problem using Law of Iterated Expectation. Got wrong answer
I'm watching MIT Intro to Probability course. And I'm stuck at a problem of Poisson process:
Lecture video in question: https://www.youtube.com/watch?v=MvGuBQZZuLM
Problem:
We have a fisherman who ...
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Show that $X$ and $f(Y)$ are conditionally independent given $Y$
Let $(\Omega, \mathcal{A}, P)$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow (\Omega_1, \mathcal{A}_1)$, $Y:(\Omega, \mathcal{A})\rightarrow (\Omega_2, \mathcal{A}_2)$ and $f:(\...
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Sensitivity Analysis with percentage-type output
I'm trying to calibrate a model, but I wanted to take a step back and perform a sensitivity analysis to assess if the parameters I'm calibrating are actually influencing the output.
The simplest ...
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How does uniform weak convergence of an empirical process carry to probability bounds at an estimated parameter?
Let $ \mathcal{F} = \{ f_\theta : \theta \in \Theta \} $ be a class of functions indexed by $ \theta $. If $ \mathcal{F} $ is a Donsker class, then the empirical process $ \mathbb{G}_n(f_\theta) = \...
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How to calculate the distribution of an average (I don't have the joint distribution but know they are strongly correlated)
The background
Admission to certain schools depends on the results of two standardised tests. The scores range from 60 to 150 in each test.
For each student, the average score across the two tests is ...
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Density Estimation?
I'm not sure if the following question of mine sound silly. I thought I would just go ahead and ask. The question is the following. We often find in probability text books questions, for example, of ...
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Teaching Poisson approximation to Binomial still relevant?
Convergence of Binomial to Poisson:
If $X_n\sim \text{Bin}(m_n,p_n)$ and if $m_n\to\infty$ and $p_n\to 0$ such that $m_np_n\to\lambda$, then $X_n\stackrel{d}{\to}\text{Poi}(\lambda)$
The above result ...
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Probability that the range of a random sample of size $n$ from a continuous distribution includes the population mean?
For the median ($m$), there is a simple formula that it falls within the range of a random sample of size $n$ from a continuous distribution:
$$
\mathbb{P}(X_{(1)} \leqslant m \leqslant X_{(n)}) = 2^{...
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Asymptotic Distribution of Weighted Empirical Distribution
Suppose that we observe an i.i.d. sample $(X_1, Y_i), ..., (X_n, Y_n)$ from $(X, Y)$. We assume that $X_i$ is bounded by $B$ and $E(X) = 0$. For some $\tau \in (0, 1)$, define the $\tau$th quantile of ...
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The simulation of the sampling distribution of a statistic, and the graph about the distribution of all the possible values the statistic can take
My questions are at the bottom. My objective is to answer these questions.
I feel that my understanding of the sampling probability distribution of a statistic is erroneous.
Definition
The Sampling ...
user493249
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Confidence intervals for predictions in ggeffects are outside the possible range of probabilities
I ran this lognormal hurdle GLMM using the R package glmmTMB:
...
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Why Is P(B∣C)=P(B) in this Bayesian Network despite an unblocked path through a collider?
When calculating the $P(E \mid C)$ for the Bayesian Network above, I got the following:
$$
\begin{align}
\mathbf{P}(E \mid C)
&= \sum_{B} \mathbf{P}(E, B \mid C) \\
&= \sum_{B} \mathbf{P}(E \...
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In a stationary time series what is the meaning of "stationary"? [duplicate]
So Here is the defintion of Stationary Time Series:
A Stationary time series is a time series whose statistical properties do not change over time. My first question is what is over time actually mean?...
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Optimal policy under RL as inference framework
In RL as inference framework (Levine, 2018) and application of language modeling, in particular seq2seq modeling, we care about learning a policy $p_\theta(y \mid x)$, where $x$ is the input sequence ...
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What is meant by a sample space to a power?
I’m reading this page:
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/13%3A_Games_of_Chance/13.03%...
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How does the sampling distribution determine the probability that this statistic falls within a certain distance of the corresponding parameter?
Problem
How does the sampling probability distribution of a statistic determine the probability that this statistic falls within a certain distance of the corresponding parameter?
Definitions
A ...
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Trying to understand random variables using the function notation f(x) = y [duplicate]
Problem
I am trying to build an intuitive understanding of random variables by expressing them in the form of a function: f(x) = y. I know this isn't the conventional way of expressing a random ...
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Is there a formal threshold for when a variable is considered 'random' in statistics? What is it?
Problem
I’ve been studying the concept of a random variable, and I have come up with this understanding of what a random variable is:
“A random variable is a variable whose values are not known nor ...
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If a biased coin is flipped once and lands heads, what is the most likely number of heads in 100 future flips?
If it is assumed that all possible probabilities of landing heads are equally likely, then the prior probability density of the probability of heads x is q(x) = 1. Bayes’ rule and the binomial ...
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if I toss a coin twice, do I have a single random variable sampled twice independently or two independent random variables sampled once? [duplicate]
This question arose in the context of CLT, but more general answers are also warmly welcome.
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Negations and Negative Probability [closed]
Talking about Negative Probability with AI
Here's an easy thought.
The author, I, who produced this document have taken undergraduate level of Introduction of Probability. Right now still in ...
