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 $Cov(X,Y)=E(Cov(X,Y|Z))+Cov(E(X|Z),E(Y|Z))$?
Amirhossein's user avatar
5 votes
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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 ...
Feri's user avatar
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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{...
Uk rain troll's user avatar
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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 ...
Carl's user avatar
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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! ...
Mike Battaglia's user avatar
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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 ...
user671269's user avatar
3 votes
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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 $p_1$ ...
Uk rain troll's user avatar
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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 ...
Yaroslav Bulatov's user avatar
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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 ...
Bhutum Banerjee's user avatar
6 votes
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158 views

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 ...
Uk rain troll's user avatar
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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 ...
Uk rain troll's user avatar
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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 ...
wuhanichina's user avatar
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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 ...
Flo's user avatar
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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 ...
Ria's user avatar
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4 votes
1 answer
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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" ...
niobium's user avatar
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7 votes
3 answers
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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(...
TheForeverFool's user avatar
4 votes
1 answer
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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 ...
Sagnik Taraphdar's user avatar
3 votes
1 answer
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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 ...
Uk rain troll's user avatar
2 votes
1 answer
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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 $\...
Uk rain troll's user avatar
6 votes
2 answers
320 views

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 ...
TheForeverFool's user avatar
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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 ...
Daniel Griscom's user avatar
1 vote
2 answers
99 views

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(...
AgnostMystic's user avatar
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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 ...
Luke Jenner's user avatar
3 votes
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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 ...
Xero0808's user avatar
8 votes
1 answer
603 views

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, ...
synack's user avatar
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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}}...
Hammed's user avatar
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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 ...
Quinten's user avatar
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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 $$A\implies B$$ $$ A$$ $$\therefore B$$ ...
niobium's user avatar
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5 votes
1 answer
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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:...
Newbie's user avatar
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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%...
JasonH's user avatar
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6 votes
2 answers
149 views

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 ...
coolname11's user avatar
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0 answers
23 views

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 ...
python noob's user avatar
2 votes
2 answers
114 views

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 ...
Dave's user avatar
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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 ...
eMathHelp's user avatar
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2 votes
2 answers
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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 ...
Quorthon's user avatar
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1 vote
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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:\, ...
user553010's user avatar
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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 ...
Wei-Cheng Lee's user avatar
1 vote
1 answer
55 views

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: ...
mafiaenshevnspiller's user avatar
2 votes
1 answer
45 views

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.
Fred Smith's user avatar
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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 ...
maldini1990's user avatar
4 votes
4 answers
267 views

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. $$e_{i+1}\leftarrow e_i - a (x_i e_i)$$ ...
Yaroslav Bulatov's user avatar
6 votes
3 answers
2k views

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....
quantrader23's user avatar
0 votes
2 answers
67 views

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 ...
Zahid Hasan's user avatar
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0 answers
30 views

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)} \...
A_Mondial's user avatar