Questions tagged [probability]

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

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Computing conditional distribution of hidden state given observed states?

I am interested in the following Gaussian linear system that describes a Hidden Markov Model (HMM): $$x_{k+1}=Ax_k + u_k + \xi_k, \xi_k \sim N_2((0,0), 0.01I_2)\\ y_{k+1}=C^tx_{k+1}+\eta_k, \eta_k \...
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Betting on a sample from a known distribution

This was an interview question. Given a known distribution, sample a value from it with replacement for many times. Two people A and B bet on the sample with their own guesses, and the one closer to ...
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"Consensus" on Analyzing Mixed Continuous and Categorical Data in the field of Statistics? [closed]

I have been trying to determine the popular "consensus" as to how mixed continuous and categorical data (e.g. a dataset that has variables on income and gender) is generally analyzed in the ...
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MNIST with a TWIST, no labels given, only probabilities

Let's say we have basic MNIST dataset, and we have the same goal to predict the digit, BUT we're swapping all the labels by RED ...
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Generate a random covariance matrix with specified eigenspectra and diagonal elements?

I want to generate a random covariance matrix ($c \in \mathcal{R}^{n \times n} $) whose eigenspectra, i.e., $n$ eigenvalues $e_0 \in \mathcal{R}^{n\times 1}$ and diagonal elements $c_{ii} \,\, i=1 \,\,...
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What is Galton's paradox?

From One Thousand Exercises in Probability I found the exercise: Galton's paradox. You flip three fair coins. At least two are alike, and it is an even chance that the third is a head or a tail. ...
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Probability of seeing sun rise tomorrow using Bayes Theorem

It is a stormy night and an alien prisoner is abandoned on earth during the first ice age. Scared, he finds a cave and only comes out for a few hours at night to hunt some food. He has no luck though. ...
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What is the correct definition of the correlation of two random variables?

In the book Signal Processing for Communications by Prandoni and Vetterli, correlation of two variables is defined as the expected value of their product, while the covariance is refered to as the &...
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Expectation values [closed]

What is the expectation value of the expectation value. How to calculate: <∆j> where, ∆j=spread of data= j-<j.> ;for any variable j
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3 votes
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What is meaning of such notations in general?

I read a notation on a paper about statistics and machine learning ("High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models" by Tengyuan Liang, Subhabrata Sen, and Pragya ...
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Probability of selecting 3 students being in different classes

I have come across this problem and I can't quite figure out the solution. A class of 60 students is divided into 3 groups of 20 randomly. Three friends are in this year group. Find the probability ...
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Do the variables used in computing the mean and variance have to be i.i.d.?

For example if you have a theoretical server that processes incoming requests where in this case there are so many requests that the callback delays depend upon each other, can you just take the ...
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In a game with 0.01 chance of survival, there are 100 participants, a specific player survives twice what are the odds?

Sorry ahead if my question is very beginner-ish but I'm confused with this example. Your help is appreciated Let's say we have a survival game where 100 players participate each time and only one of ...
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Probability of bringing a card that is already in the collection

Let's say there is a card collection that is made of 1,000 different cards. I start calling my friends and each of them brings 15 random cards to the collection. The first friend will bring 100% new ...
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Uniform distribution of skewed dice and the central limit theorem

I am reading MacKay's Information Theory, Inference, and Learning Algorithms. I am not a student. On pg. 36, ex 2.16, c and d: C. How can two cubical dice be labelled using the numbers {0,1,2,3,4,5,6}...
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If Y is the sum of normalized multivariate pareto random variables then Y is a Feller-Pareto random variable

If we let $\underset{k\times 1}{\boldsymbol{X}}=(X_1, \dots, X_k)' \sim MP^{(k)}(\boldsymbol{0},\boldsymbol{1}, \alpha)$ where MP denotes a Multivariate-Pareto distribution, with joint survival ...
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Issue with bounded in probability

I have tried to prove the following problem that I read in the lecture and it seems not transparent to me. Suppose that $Y_{i}$ be independent random variables (with $i=1,2,3, \dotsc$). Each has the ...
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3 votes
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Probability for a bin in a binned histogram

This question is very basic, but I cannot figure the error in my thinking. According to the author of the book "Pattern Recognition and Machine Learning", we can get the Probability ...
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Fitting discrete data to continuous distributions

I'm creating a simulation model, in which some stochastic factors are included. On of my stochastic factors is the amount of containers arriving daily for a specific delivery location. A plot of this ...
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Given two sequences of random variables whose moments match, does their difference tends to 0?

Suppose you have two sequences of random variables $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ and you know that for every $n$ $$ \mathbb{E}[X_n^r] = \mathbb{E}[Y_n^r] \quad \forall \, r ...
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Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent

Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$. Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=...
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How do I calculate the constant K that makes the area under the pdf equal to 1 [closed]

You are analyzing the probability of defective machine parts coming out of an assembly line. Data collected over 60 consecutive days, revealed 2 days where defects were observed in machine parts. To ...
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How many random draws needed to sample a uniform space?

I am looking for a way to estimate how many random draws took place to produce a produce a known number of unique results. I'm sampling a uniform distribution. For example, if there were a lottery ...
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Logistic regression simulation with respect to event occurrence (prevalence)

I am trying to simulate logistic regression data, but under the constraints of prevalence. $$\text{logit}(y_i) = \beta_0 + \beta_1 X_1 + \beta_2X_2$$ For example, I want to create a dataset that has ...
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Consecutive coin flips, what is the appropriate statistical test for this word problem? [closed]

I was listening to a podcast by NDGT (Neil deGrasse Tyson, a prominent scientist) and he posed a simple thought experiment to illustrate the susceptibilities to cognitive bias. What I've come here to ...
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2 votes
1 answer
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Prove $P(X_1+X_2> 2C) \leq P(X_1>C)$ if $X_1,X_2$ are identical, but dependent?

If $X_1,X_2$ are dependent but identically distributed, it seems obvious that $P(X_1+X_2\geq2C) \leq P(X_1\geq C)=P(X_2\geq C)$. At least if we additionally assume that the joint distribution is ...
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Multivariate analysis of sample mean and sample variance

$\{X_n\}$ Let be a sequence of iid probability vectors with mean vector$ \mu$ and variance-covariance matrix$ Σ$. In this case, sample variance and sample covariance are defined as follows $S_{n,j}^2=\...
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Distribution function of the linear combination of standardized student-t quantiles [duplicate]

Assume to observe 2 quantiles, x and y, associated with the z% probability. These quantiles are generated by 2 non-independent standardized student-t distributions X and Y. In case of linear ...
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When I have population data do I need probability and can I infere? [duplicate]

Say I have data about a population and I want to analyze them, then in my understanding the "sample" is the population. And then do I need probability, like p-values, confident intervalls ...
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Calculating Probability Using Bayes? [closed]

A recent survey of residents in Texas concluded that 55% of Austin city residents and 46% of Houston city residents broke a bone at some point during their childhood. Let’s say Austin has 5200 ...
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Fitting Sparsed Constrained regression with non-negative coefficients adding to 1

I see a similar problem in How do I fit a constrained regression in R so that coefficients total = 1? Specifically, my model is $Y_i= \pi_1 X_1+\pi_2 X_2 +...+ \pi_K X_K +\epsilon_i$ with $\pi_k \ge 0$...
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2 votes
1 answer
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probability that the players will exchange their initially drawn number

Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own ...
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Notation of expectation with conditional in subscript

Inside the book "The Elements of statistical learning", I stumbled upon the following notation (Ex. 2.7) $$E_{\mathcal{Y|X}}(f(x_0) - \hat{f}(x_0))^2$$ where $\mathcal{X, Y}$ are two random ...
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What is the intuition behind the odds scale?

What is an intuitive explanation of the odds scale? In a logistic regression such as $$logit(p) = \beta_0 + \beta_1 x$$ we often interpret $\beta_1$ by looking at the odds ratio, $e^{\beta_1}$, which ...
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How to calculate real probability with multiple scenarios?

I would like to know how to calculate the probability of being profitable on an event where there are multiple scenarios. For example paying 5 dollars to grab a bill from a box box full of bills where ...
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Conditioning of join gaussian over a line

I need to compute the conditional probability of bivariate normal distribution over a line. Let's suppose that X and Y both are normal distributions and that they are independent. Let's suppose that ...
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What is the probability mass function of Rock, Paper, Scissors?

I was curious about the statistics behind the game of rock, paper, scissors. Let's say n people are playing, where n is greater than or equal to 2. If when all n people reveal their play and only 1 ...
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General way to describe dot product of N different random variables [duplicate]

I'm a novice to DS, so feel free to correct me. Imagine we have $N$ biased coins each with different probability of getting heads (which are known to us prior). What's the probability of getting $k$ ...
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Conditional probability problem (Bayes Theorem ?) [closed]

Hello, basically, I can't find 1/2 for the very last question. I tried to use the Baye's Theorem, but it wasn't successful Could someone help me ?
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Given 15 random birthdays in a year, what are the odds that there are at least 120 consecutive days in a year without a birthday? [closed]

The 120 days can be shared between consecutive years too, such as roughly November through February.
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1 vote
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MLE of the Uniform Distribution

In a uniform distribution where $0\leq X \leq \theta$, the pdf is represented as $f(X|\theta) = \frac{1}{\theta}I(0\leq X \leq \theta)$, and the likelihood is $L(\theta) = \prod\frac{1}{\theta}I(0\leq ...
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MLE for the Uniform distribution [duplicate]

I understand how a random sample $x_1, ..., x_n$ following the Uniform Distribution with $0 \leq x \leq \theta$ has a log-likelihood proportional to $\frac{1}{\theta^2}$. I am told that the MLE for $\...
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Do Mixture Models "Defy" Entropy?

Recently, I have learned about the principle of Maximum Entropy with regards to Probability Distribution (https://www.youtube.com/watch?v=2gTrsLVnp9c) - in particular, when certain "information&...
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Expected value of a random variable with truncation

Let $f:[0,\infty)\to \mathbb R_+$ denote the PDF of a random variable $X$ and $c>0$ a constant. I want to evaluate the following integral: $$I(c)=\int_0^\infty{\min(x,c)f(x)dx}.$$ This can be ...
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Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability?

...
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How to actually apply martingales when conditioning on a random variable (not filtration)?

For a Galton-Watson process, I've shown that $\frac{Z_n}{\mu^n}$ is a martingale i.e. $E[\frac{Z_{n+1}}{\mu^{n+1}}|\mathcal{F}_n]=\frac{Z_n}{\mu^n}$. However, I want to show that, for $n>m$, $$E[Z_{...
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Finding probability of a team winning the title given probability of future matches

Say, we have two teams: Manchester City and Liverpool. Both of them have 3 games remaining with the following probabilities of winning/drawing: ...
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What is the conditional expectation of two random Poisson variables?

Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
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T-testing on sparse data

Problem: I have two stochastic processes, $S_1$ and $S_2$, that frequently are zero, but occasionally have positive values with unknown probabilities $q_1$ and $q_2$. e.g. $$ S_1 = \{0,0,0,0,0,21,0,0,...
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3 votes
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Calculating the probability of guessing 3 card draws out of 6

My friend guessed the correct card drawn from a full deck of 52, 3 times out of 6 guesses (card not replaced each time). I'm trying to work out the probability. My thinking is that if you guessed ...
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