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Questions tagged [probability]

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

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Notation of probability distribution [closed]

Does $P(X)$ mean the probability distribution of random variable X? Sorry, I can't find the right resource on Google.
William Zhao's user avatar
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3 answers
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How to show that many functions (a hundred, a thousand) have the same shape an distribution of values over an interval?

I have functions that on iterval [0,1] all seem to look like this: i.e. they have a zero around 0.4 +ve derivative from zero to 0.4 and around zero or slightly negative derivative up to 1. I plan to ...
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1 answer
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Are all random variables estimators? [duplicate]

My hand-wavey understanding is a random variable is a function from a domain of possible outcomes in a sample space to a measurable space valued in real numbers. We might denote a random variable from ...
Estimate the estimators's user avatar
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Proof of Strong consistency of Beta posterior distribution

Suppose that we have random variable $X_{1}, X_{2}, ..., X_{n} \sim^{iid} \text{Bernoulli}(p_{0})$ with $p_{0}$ true unknown probability in $[0,1]$. Now, I want to implement Bayesian machinery to ...
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Probability of fixing error in application, given successful and unsuccessful runs before fix

An application communicating with a database started to fail with network errors. The errors occur only during some runs, and at random times when running the application. If a network error occurs, ...
sebastian's user avatar
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16 votes
6 answers
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Equivalence of first/second choice with naive probability - I don't buy it

I'm seeking a better understanding of the following problem from Blitzstein and Huang (2015) (Chapter 1, Exercise 31, p. 35): A jar contains $r$ red balls and $g$ green balls, where $r$ and $g$ are ...
Trent's user avatar
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1 vote
1 answer
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Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? [closed]

Is the variance of the mean of a set of possibly dependent random variables less than or equal to the average of their respective variances? Mathematically, given random variables $X_1, X_2, ..., X_n$ ...
HappyFace's user avatar
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1 answer
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Bayes' Theorem applied in real study [closed]

My name is Molly, and I am a medical student at Queens. I'm delving into the realm of statistics. It's essential to comprehend how statistical methods are utilized in medical screening and diagnostics....
Shuangshuang Li's user avatar
5 votes
1 answer
129 views

Equivalence of inverse transformations under distributional equivalence

Consider continuous, invertible transformations $g,h : \Bbb{R}^d \rightarrow \Bbb{R}^d$ and suppose $g(Y) \overset{d}{=} h(Y)$, where $Y$ is a $N(0, I)$ random variable. Then what can we infer about ...
ArunavB's user avatar
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relating correlation to probabilities

Given two centered and scaled random variables $X$ and $Y$, can you relate the probability they have the same sign to their correlation? If the correlation is close to $1$, I am picturing the joint ...
kara890's user avatar
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2 votes
1 answer
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Let $X(t)$ be a Gaussian process. Does $\mathbb{E}[X(t)^2 X(s)^2] = \mathbb{E}[X(t)^2 ] \mathbb{E}[X(s)^2 ] + 2 (\mathbb{E}[X(t) X(s)])^2 $?

As the title says, can I apply Isserlis' theorem to $\mathbb{E}[X(t)X(t)X(s)X(s)]$?
hipHopMetropolisHastings's user avatar
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100 prisoners problem with n < 50

In the 100 prisoners problem, I understand that when $l \geq 51$, the number of permutations is $100!/l$. However how to calculate the exact number of permutations when $l < 51$? When $l = 1$, the ...
Green's user avatar
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0 answers
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Bayesian network with partial info (4 nodes)

I have some (conditional) probabilities for a Bayesian network with binary variables, but not all. My DAG is M->F->Y->C<-F and M->Y and M->C I ...
Dirk N's user avatar
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2 answers
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If the variance converges to zero, when do we have almost sure convergence

We have that $\mathbf{E}(X_n)=c$ where c is a positive constant and $\lim_{n \rightarrow \infty} \mathtt{Var}(X_n) =0$. Then $$ X_n \rightarrow c \quad \mbox{in probability as} \quad n \rightarrow \...
GCru's user avatar
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3 votes
1 answer
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Evaluating the Validity of Election Results

In a suspicious election involving 5 candidates, the announced votes for each candidate are a multiple of 3. The chance of all five numbers being a multiple of 3 is only 0.41%. Therefore, some people ...
Amin's user avatar
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2 votes
0 answers
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Complex parameterizations of real-valued distributions

Suppose we have some random variable $X$ that takes values in $\mathbb{R}^n$, parameterized by $\theta \in \Theta$ where the parameter space $\Theta$ is finite-dimensional. In almost all statistical ...
Randy Savage's user avatar
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Clarifying the Use of Binomial Distribution vs. Bayesian Theorem in Bridge

I am exploring probability concepts as they apply to the game of bridge and am seeking some clarification on when to use binomial distribution and when to apply Bayesian theorem. My understanding is ...
Shuangshuang Li's user avatar
0 votes
1 answer
46 views

Calculating the joint pdf of linearly dependent random variables $X$ and $Y=X$

Let $X$ and $Y$ be two random variables and $p_{(X,Y)}(x,y)$ be the joint pdf of $(X,Y)$. Suppose that $(X,Y)$ transformed to $(X,X)$. We want to calculate the joint pdf of transformed random ...
Naveen Kumar's user avatar
3 votes
1 answer
102 views

Applying Bayesian probability to a generalized Monty Hall problem

I posted this question about the Monty Hall problem and Monty's knowledge of the probability distribution several months ago. I got some good answers and this one in particular helped me gain some ...
Mikayla Eckel Cifrese's user avatar
2 votes
0 answers
42 views

Integral of stochastic processes

Suppose that i have a random variable $I(t) = \int_0^{t} N(s) e^{\sigma W(s)} ds$ where $N(s)$ is the number of arrivals at the time s ( notice that is not the total arrivals until time s, just the ...
Bruno Llacer trotti's user avatar
4 votes
3 answers
154 views

Is $F_X(t) > F_Y(t)$ a sufficient and necessary condition of $\mathbb{P}(X < Y) > 0.5$

I asked the question when conducting Mann–Whitney U test using the scipy implementation. The null hypothesis for one of the one-sided tests is $F_X(t) > F_Y(t)$, where $F$ denotes a CDF. I wonder ...
zyxue's user avatar
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Gambler ruin's: Probability of k consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
Zhihao Xu's user avatar
1 vote
0 answers
33 views

Is it possible to compare the output probabilities of two machine learning models? [closed]

Let's suppose I have two classification machine learning models: $\text{Model}_1$ and $\text{Model}_2$. Each of them are not necessarily the same algorithm, and have not been trained necessarily with ...
Poisson Parade's user avatar
2 votes
3 answers
97 views

How to work out the expected rate of success when there is a guaranteed success on the nth attempt?

I'm looking to work out how to find the expected success rate when given the rate of success but, also after n-1 failed attempts, there the success rate is 100% for the nth attempt. Intuitively I ...
BLey99's user avatar
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1 vote
0 answers
19 views

central moments of random variable from _estimates_ of draws from the distribution function

I am trying to estimate the first two central moments of random variable $r$. The information I have about $r$ is a set of estimates $\hat{r}_i$ for $i \in \mathcal{I}$, each with corresponding ...
daydaybroskii's user avatar
0 votes
1 answer
29 views

Bayesian updating with affine transformation of random variable

I want to estimate a parameter $\theta$, and I have a prior $\pi(\theta)$. I receive the realization of a random variable $Y$, which has some likelihood $f_Y (y \mid \theta)$. My posterior then ...
Joao Francisco Cabral Perez's user avatar
11 votes
2 answers
416 views

(THEORY) Do Tree models output probabilities?

I have a question purely theoretical about decision trees outputs for classification. I have heard a lot of people say "the output of tree models are not probabilities", and having studied ...
Felipe Araya Olea's user avatar
5 votes
2 answers
132 views

Do tail bounds on probability translate into bounds on expectations?

Suppose I have a bound of the form: $$P(X \geq t) \leq \exp(-t^2).$$ Can I say anything about the expectation of $X$, $E[X]$? In particular, can I get a bound on $E[X]$? Here's the specific case that ...
snickerdoodles777's user avatar
2 votes
1 answer
41 views

How to evaluate whether the results of multiple games conform to the probability distribution of their respective states

There are $n$ games, and each game has a state after the start. Based on this state, the probability of the outcome of the game can be calculated. For example, there are 3 type of results, and the ...
keook monsk's user avatar
4 votes
3 answers
92 views

Posterior expectation of normal distribution with "truncated" observation

Consider the following problem of estimating an unknown parameter from normal samples: Suppose that $\theta \sim N(0, \tau_\theta^{-1})$, where $\tau_\theta \ge 0$ is the prior precision. Consider two ...
keepfrog's user avatar
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0 answers
54 views

Question about CDFs

Suppose that I have two arbitrary random variables $Y,X$ with strictly increasing CDF. Let $\omega(X)$ be a known function of $X$ that is known in advance and $\mu$ be a scalar parameter. Assume that ...
ecnmetrician's user avatar
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0 answers
18 views

Transformer model conditional probability distribution of sub-sentences

I have a simple transformer model (decoder only) which is trained on some dataset containing sentences to do next-word prediction. The model captures a probability distribution $P_{\theta}(\mathbf{a})$...
JazzJammer's user avatar
4 votes
1 answer
227 views

Approximation function for MLP and LSTM

I have total of 6300 samples, 5800 of which are training data, and 500 of which are testing data. We compare the performance of LSTM and multilayer perceptron (MLP) with one hidden layer in terms of ...
D. S.'s user avatar
  • 69
0 votes
0 answers
17 views

Finding a function for the number of people on a bus

I've been trying to pin down the function that would describe the number of people on an ideal bus ( # of stops and passengers approaches infinity/ is large enough that the function is continuous), ...
Eda Toloch's user avatar
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0 answers
45 views

Help with completing a derivation of usefulness of cross-validation

This question is raised as a result of my attempt to answer this other question of mine. Let's refer to all our prior knowledge, both explicit and implicit, as $X_\text{true}$. Almost always, we are ...
Feri's user avatar
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0 votes
0 answers
23 views

How to Compute the Posterior Distribution of Covariance matrix in a Matrix Normal Model with Inverse Wishart Prior

I am working on a time series model involving Kalman filters and smoothing to estimate state variables $Y_i$. The part of model is structured as follows: $Y_1, \ldots, Y_n$ are iid. $Y_i \sim \mathcal{...
Ayden Frost's user avatar
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0 answers
21 views

Computing a Confidence Interval for E[X] when PMF is given

I am given a Probability Mass Function for a discrete random variable. From the PMF I computed the Expected Value $E[X]$, the Variance $V[X]$ and the Standard Deviation $S[X]$. Here is an example (the ...
rusiano's user avatar
  • 566
15 votes
1 answer
487 views

What distribution should I use to predict three possible outcomes

I am 70, left school at 14 but took to maths a few years back to ward off dementia so please excuse the naivety of my question. I have been using Poisson distribution to solve my problem but I dont ...
Simon Bates's user avatar
1 vote
0 answers
124 views

$E[X\mid U,V]$ with dependent random variables [closed]

Let $(X,Y,Z)$ be a three-dimensional random variable with density function $f(x,y,z)=\frac{2}{3}(x+y+z)$ on $0<x,y,z<1$, $U=\min(X,Y,Z)$, $V=\max(X,Y,Z)$. Calculate $E[X\mid U,V]$. I think the ...
Speltzu's user avatar
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1 vote
0 answers
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Probability of leaving with $20

You play a game with a coin. You may place a bet; if Heads is flipped then you receive your bet back plus the same in winnings. If Tails is flipped then you lose your bet. You have 20 dollars and you ...
Yuna's user avatar
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0 votes
0 answers
25 views

Expected value of a decreasing function of two random variables

My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
irodr's user avatar
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1 vote
0 answers
11 views

Estimating Markov Chain Probabilities with Limited Data

Suppose I have some data on transitions between states of a Discrete Time Markov Chain. Let's say that transitions between some events are observed more frequently from others. For example, in a 3 ...
user avatar
0 votes
0 answers
32 views

What is the difference between estimating parameters via MLE versus minimizing deviations from expectation?

What is the difference between estimating parameters using MLE (or MAP with uniform priors): $$\theta^* = \arg \max_\theta p(X|\theta)$$ and estimating them according to which setting would engender ...
actinidia's user avatar
  • 145
9 votes
2 answers
518 views

Markov Chains with Changing Number of States

I have seen these kinds of Discrete State Markov Chains before (Continuous Time or Discrete Time): Homogeneous (Probability Transition Matrix is constant) Non-Homogeneous (Probability Transition ...
udon762's user avatar
  • 91
0 votes
0 answers
20 views

Data generation processes that "vary slowly enough"

I am considering some abstract data generation processes producing data by first drawing features $x_{1:n}\sim p(x)$ iid, and then drawing the responses $y_i\sim p(y\mid x_i)$ iid, where $p(y\mid x)$ ...
Daniel Robert-Nicoud's user avatar
4 votes
1 answer
187 views

$E[X\mid \max(X,Y,Z)]$ with dependent variables

Let $(X,Y,Z)$ be a three-dimensional random variable with density function $f(x,y,z)=\frac{2}{3}(x+y+z)$ on $0<x,y,z<1$. Calculate $E[X\mid \max(X,Y,Z)]$. I think the answer is $\frac{25}{36}\...
Speltzu's user avatar
  • 336
7 votes
4 answers
267 views

Estimating Probability Density for Sample

I have a dataset of over 20,000+ samples. The objective here is to define a distribution for the sample so that I can plot all possible outcomes. However, I am unable to find an appropriate ...
Ahmed Jyad's user avatar
0 votes
0 answers
21 views

Doubt regarding limiting distribution on Vasicek model

I was reading an article from Vasicek where he's concerned about deriving the limiting loss probability distribution on a credit risk model with 2 factors. I am here presenting a somewhat different ...
Chaos's user avatar
  • 431
0 votes
1 answer
34 views

Can two different measures have the same first order stochastic dominance?

$(S,\Sigma,\mu)$ is the common probability triple, where $S=[0,1]$, $\Sigma$ is the Borel sigma algebra, and $\mu$ is the Lebesgue measure. $X:[0,1]\to\mathbb R$ is a r.v. There are two different ...
dodo's user avatar
  • 185
0 votes
0 answers
23 views

Poker strategy with blind betting

I came across a version of Texas hold ‘em poker where if you’re blind, your chip value is doubled at betting time (ie. to call 10, you only put in 5). You can look at your cards at any time, at which ...
olivarb's user avatar
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