Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
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Prediction Machines Word Problem (Conditional Probability?)
I am new to statistics but had what I think is a pretty simple question:
Prediction machine 1 correctly guesses the outcome of binary (yes/no) events 60.4% of the time.
Prediction machine 2 correctly ...
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Theorem 5.1.4 from High-Dimensional Probability - Concentration on the Sphere [duplicate]
Let $M$ denote the median of a Lipschitz function $f(X)$ with Lipschitz norm equal to 1. How can I show that if $\left \| f(X)-M \right \|_{\psi_{2}}\leq C $, then $\left \| f(X)-\mathbb{E}f(X) \right ...
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Prove that the equality holds [closed]
How to prove that for any random variables $X$, $Y$ and $Z$ with finite variances, we have $Cov(X,Y)=E(Cov(X,Y|Z))+Cov(E(X|Z),E(Y|Z))$?
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A seeming paradox regarding estimation of the number of buttons
There is a computer with $N$ buttons in a secret room. We do not have access to the computer and we do not know $N$. But we know that $N\leq 100$ and we have a ever so slightly larger prior for ...
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How to test the statistical significance of probabilities?
I am working on a model with as output a list of probabilities. If I have two lists of such probabilities, how can I test if the difference between the two lists is large?
Thanks!
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Is it possible to Discretize a Continuous Time Markov Chain?
In a Continuous Time Markov Chain (CTMC), the following properties are said to hold:
Discrete (Embedded Jump Process):
$$P_{ij} = \frac{q_{ij}}{\sum_{i} q_{ij}}$$
$$q_{ij} = \lim_{{h \to 0}} \frac{...
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]
Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
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The Bing Tibetan Glitch Emoji Problem
Here's a very interesting mathematical statistics puzzle that I randomly stumbled into while using Bing. This turned out to be deep enough that I spent quite some time thinking how one could solve it! ...
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Converting an integral into a probability of some event
Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$
$$$$Consider the ...
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Writing Maximum Likelihood Equations for a (Hidden) Coin Toss Problem
I posted this question here Estimating Coin Flip Probabilities with Missing Information today.
Suppose there is a game where there are 2 Coins (Hidden Markov situation):
Coin 1 has a $p_1$ ...
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Efficiency of chi-squared denoising
Suppose my measurement $\theta+\epsilon$ is corrupted by IID additive noise $\epsilon$ distributed as chi-squared with (known) $d$ degrees of freedom, what is the efficiency of pooling multiple ...
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Is a sub Markov chain also a Markov chain?
Let us assume $A \rightarrow B \rightarrow C \rightarrow D$ is a markov chain. Can we also state that $A \rightarrow C \rightarrow D$ is also a Markov chain? It intuitively feels right. Can anyone ...
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Estimating Coin Flip Probabilities with Missing Information
I am trying to create an example that shows how the quality of estimation is impacted by incomplete information (e.g. deliberately neglecting the Markov Property) and wrong assumptions about the data ...
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Why is the Acceptance-Rejection Sampling Algorithm able to produce samples from the target distribution? [duplicate]
I am trying to understand why is the Acceptance-Rejection Sampling Algorithm able to produce samples from the target distribution. I have seen several explanations for this, but have never been able ...
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Maths Questions - Probability distribution of choices
I need help with a question I am trying to work out:
Individuals have 5 choices:
Choice 1: utility1 = alpha * R1 + beta * C1 + random_shock_1(mu=0,sig=1)
Choice 2: utility2 = alpha * R2 + beta * C2 + ...
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Estimation of a generalized noncentral chi-square dsitribution
According to Mathai A, Provost S: Quadratic Forms in Random Variables: Theory and Applications, 1992, the quadratic form of normal variables has a generalized noncentral chi-square distribution (or ...
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Formula to find percentage of winning between two different numbered dice
Lets say I have a dice with the numbers from 1 to 10 and another one numbered from 1 to 7. I roll both dices once. Whats the chances in percentage that the 10 numbbered dice will win in terms of ...
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Probability that team A will win overall match
Team A and Team B are competing in a sports game and the score is currently tied at 10-10. The first team to win by a margin or two will win the tournament. Team A has 65% chance of winning each point ...
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Probability Theory (Jaynes' approach based on Cox theorem) CDF how to prove increasing monotonicity of this function?
In his book Probability Theory: the Logic of Science, Jaynes defines at page $107$ (chapter $4.5$) the continuous probability function.
To do so, he introduces a real continuous "random" ...
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The probability of an event is the sum of the probabilities of its sample points
Let $S = \{s_{1}, ..., s_{n}\}$ be a finite set. Let $\mathcal{B}$ be any sigma algebra of subsets of $S$. Let $p_{1}, ..., p_{n}$ be nonnegative numbers that sum to 1. For any $A \in \mathcal{B}$, $P(...
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Why is this derivation of the mean of the gamma distribution using the log-partition function incorrect?
I am using this formulation of the exponential family :
$$
\large f_{X}(x;\boldsymbol{\eta})=h(x) \exp \left(\boldsymbol{\eta} \cdot \mathbf{T}(x)-A(\boldsymbol{\eta})\right)
$$
The gamma distribution ...
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Why are Likelihood Functions from Exponential Family Convex?
I am trying to understand why the Likelihood Function of any distribution from the Exponential Family is Convex. This convexity property makes optimization/parameter estimation easier when working ...
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Do "Likelihood Properties" apply to "Non-Likelihood Solutions"?
If $X$ is a random variable with PDF $f(x;\Theta)$, then we define the likelihood function as:
$$L(\Theta; x) = f(x;\Theta)$$
As I understand, we say that $\Theta^*$ is the true MLE estimator if $\...
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How to define the domain and codomain for the probability function of an uncountable sample space?
The domain and codomain of a probability function for a discrete sample space can be defined as ...
Domain: Power set of sample space
Codomain: $y \in [0, 1]$
How can I define the domain and codomain ...
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Real-world testing: estimating chance that there's an unknown variable/constraint/relationship
I'm working on a hardware product where reliability is paramount. So, one test I've done is to power-cycle units repeatedly, seeing if they properly start up. Result in 10k power-cycles of a group of ...
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The training error of best hypothesis
Let $\mathcal{X}$ and $\mathcal{Y}$ denote the domain set and label set respectively. Also let $\mathcal{D}$ be a distribution over $\mathcal{X}$ and $f:\mathcal{X} \to \mathcal{Y}$ be the true ...
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Density of distance from orgin in a general disc
Consider the disc $|z-c| \leq R.$For the case of a unit disc ,the probability desnity of the distance from origin of a randomly and uniformly chosen point in the disc is given by
\begin{equation}
f(...
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Applying one frailty value to all observations in mixed effects cox regression and then recalculating the survival probability
I want to test the difference in survival probabilities for if every observation in my dataset was part of the frailty group with the worst or best frailty value. I want to calculate the survival ...
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How to calculate probability of a team winning in penalty kick?
I was browsing the web and I saw this chart. I understand penalty rules but I really have no idea how to calculate the numbers in the chart, right on the first line.
My idea is to consider each shot ...
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Lack of rigor when describing prediction metrics
I constantly see metrics that measure the quality of a classifier's predictions, such as TPR, FPR, Precision, etc., being described as probabilities (see Wikipedia, for example).
As far as I know, ...
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Finding joint probability distribution of standard normal with constraints
Let $X, Y \mathop{\sim}\limits^{iid} N(0,1)$.
a) Suppose $X < Y$, find the joint pdf of $X$ and $Y$.
b) What is the joint pdf of $X$ and $Y$ if $X = Y$?
We know that $f_X(x) = \frac{1}{\sqrt{2\pi}}...
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Calculate Expected Goals (xG) using Bernoulli
I would like to calculate the Expected Goals (xG) using a Bayesian model. The xG is the probability of a shot resulting in a goal or not. The value is between 0 and 1 meaning 0 is most likely not a ...
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Conditional probability - when is the fraction defined (representation of a strong syllogism with conditional probabilities)?
In Jaynes' book Probability Theory: the Logic of Science (page $65$ in the pdf, i.e. page $35$ in the book), the author says that the strong syllogism
$$A\implies B$$
$$ A$$
$$\therefore B$$
...
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Is zero condition mean preserved after transforming the conditioned variable?
In my econometrics class regarding multiple linear regression, we learned that one of the Gauss-Markov assumptions is the zero conditional mean, expressed as $ E(y|\boldsymbol{x}) = 0$.
My question is:...
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Definition of expectation with condition variables
I am having a hard time of digesting this, which is part of EM algorithm that I borrowed Equation 3.2.7 from https://www.informit.com/articles/article.aspx?p=363730&seqNum=2#:~:text=3.2%...
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Upper Bound on $\mathbb{E}[\frac{1}{1 + X}]$ where $\mathbb{E}[X] = a$ and $0<𝑎<1$
$𝑋$ is a positive random variable (potentially unbounded) with $0 \le \mathbb{E}[X] = a < 1$.
Since $\phi(x) = \frac{1}{x}$ is a convex function, we can use Jensen's inequality to derive a lower ...
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How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
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Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
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Probability of victory in competition
I imagine a scenario where two teams play each other having historical win percentages. Team $1$ is quite good and wins $60\%$ of its games. However, team $2$ is really good and wins $90\%$ of its ...
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Find PDF from approximated MGF
I have an array of values of MGF (it is evaluated at some points).
The plot of it is shown (blue curve): .
Is it possible to find PDF knowing MGF in such form?
I tried to fit MGF with some curve (you ...
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What does the following mean with regards to Poisson distribution?
A text I'm reading says the following,
Consider the occurrence of any uncertain event over time or space in such a way that the average occurrence of the event over unit time or space is m. We may ...
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Question about the mean first passage time
A homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with discrete state space $\mathcal{S}$.
Consider the minimum number of steps to visit $k\in \mathcal{S},$
$$\tau_{k}:=\text{min} \left\{n\ge 1:\, ...
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Is a majority vote from two classifiers trained on m data each better than one classifier trained on 2m data?
How does the performance of ensemble classifiers compare to a single classifier in the PAC learning framework? Specifically, consider $2m$ i.i.d samples $(x, y) \in \mathcal{X} \times \{-1, 1\}$ from ...
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Help finding unknown distribution (for fun)
I want to estimate the expected return of three different distributions.
So there is this game I play, which has a function where you invest in a marketing campaign. There are three different types: ...
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Multiplying independent conditional probabilities
If we know A and B are independent such that $P(A,B) = P(A)P(B)$, does $P(A|C)P(B|C)=P(A,B|C)$?
Intuitively, it seems like this should be true, but I don't know how to prove this formally.
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Fine-tuning naive bayesian model for text classification (multi-categorical outcomes)
I have a dataset of thousands of reddit posts on economic issues in the U.S. We selected a random sample of 40% of total posts where each post contained a "Blame" indicator with three ...
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Divergence of $e_{i+1}\leftarrow e_i - x_i e_i$ for Cauchy $x_i$
Suppose $e_0=1$ and $e_k$ evolves according to the following recurrence with $x_i\sim \operatorname{Cauchy}$, IID draws from standard Cauchy random variable.
$$e_{i+1}\leftarrow e_i - a (x_i e_i)$$
...
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Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)
$X$ and $Y$ are independent variables $X$ ~ $N(0,2)$ and $Y$ ~ $N(0,1)$. What is the probability that $X > Y$? I understand that the distribution of $X - Y$ ~ $N(0,3)$ assuming they are independent....
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Are mutually exclusive events are independent events? [closed]
So far my understanding:
consider two mutually exclusive events, A and B. If A occurs, B cannot, and vice versa. In this case, the occurrence of one event (say, A) provides certain information about ...
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Concentration inequalities for local smoothers (Nadaraya Watson)
Let $m(X)=E(Y|X)$ be a regression function with random design and let $\hat{m}_h(x)$
\begin{equation}
\hat{m}_h(x)=\frac{\sum_{i=1}^n K_h\left(x-x_i\right) y_i}{\sum_{i=1}^n K_h\left(x-x_i\right)}
\...