Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
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Completing the square and marginalizing a multivariate Gaussian [closed]
Edit: This question has been closed for being unrelated although I see similar questions posted here with the same objective, yet not with enough detailed answers or not exactly what I am looking for (...
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Estimating the probability of a sum of events
I have n machines that use the same utility. Each machine randomly demands a unique f_n flow rate of the utility once every h_n hours on average. Each machine's demand event lasts for about m_n ...
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Basic probability question but struggling (brain teaser with friend)
You are presented with a bag of 4 orbs. You know that 2 are blue and 2 are red. You start drawing them without replacement from the bag one-by-one. Before each draw, you are given the opportunity to ...
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How many numbers can I generate and be 90% sure that there are no duplicates?
Suppose I am generating random 4-digit numbers. Obviously there are 10,000 possible numbers, but the chances are I will get a duplicate long before I generate that many.
Can anyone explain how I would ...
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Predicted Probability and Interaction Terms
This is my first time asking a question here, so apologies if I failed to include all necessary information. I have run GLMM's in the past but have limited experience with assessing interaction terms.
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How to calculate Revenue of combined listing
I have a 3-floor apartment with 6 bedrooms. I have listed it on Airbnb in 4 different ways: (1) Entire apartment with 6 bedrooms, and (2) Three separate listings for each floor, each with 2 bedrooms.
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Find the CDF of a Probability Density Function [closed]
I'm having trouble with this problem. I tried integration using -1 as the lower limit, and x as the upper limit. I got $\frac34x - \frac{x^3}3 + \frac5{12}$, but I'm getting the sense that is wrong.
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The dependency in the instances of a random variable [duplicate]
Suppose we have a list L of all root words in English numbered from 1 to n. The data is any English text (let's say a text from a book) where each word is replaced by its root. You are given the data ...
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How to compute 95% Credible Interval for a T distribution with known Mean and Variance
I have a non-central T distribution from a predictive posterior $P(x|D,\theta)=t(\mu,\tau^2,v)$ with known parameters and I want to compute the 95% credible interval of the mean prediction.
I know ...
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Conditional probabilities of a chain of variables depending on one another
Suppose there are $T$ variables $\{x_1, x_2, ..., x_T\}$, and they have the following relationship
$$
p(x_t|x_{t-1}) = N \big( x_t; \sqrt{1-\beta_t}x_{t-1}, \beta_tI \big)
$$
that means there is a ...
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Statistical model for likelihood that an athlete will "place" in track and field
I'm trying to come up with some sort of statistical model to predict how likely an athlete would be to place in a track and field event (e.g. 100m run) based on their time. I was thinking it would be ...
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Probability, Sum of RVs [closed]
Let X and Y be independent exponential random variables with expected values E[X] =
1/3 and E[Y ] = 1/2. Find the PDF of W = X + Y .
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Are there conditions for which the Pareto distribution arises? Are there characterization theorems of the Pareto distribution?
There are many real-world phenomena in which a variable of a population follows the Pareto distribution. I am wondering, what are the sufficient conditions for the distribution to be Pareto? And if it ...
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Simulating the outcome of X number of random trials
Let's say I want to flip a coin a billion times and count how many heads I get. But I don't have time to actually flip a coin a billion times.
Assuming I have a random number generator than can ...
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How to create optimal cut-off scores for a test placing students into different courses
Our goal is to determine optimal cut-off test scores for course placement. The course placement has already been manually assigned to each test-taker. The goal is to replace this manual labor with the ...
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If the integral of the product of t and phi(t) converges, does the random variable have continuous density? [migrated]
If $\int|t\varphi(t)|\mathrm{d}t < \infty$, where $\varphi(t)$ is the characteristic function of a random variable $X$, does $X$ have continuous density?
My current thinking is: not necessarily. ...
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A simple probability model for an epidemic simulation
I'm creating a epidemic simulation program in Python, but I'm stuck trying to determine the probability of being infected (if exposed).
I have an m x n grid with 0's and '1 (0's represent healty cells,...
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Bayesian Inference: Conceptual question to get evidence
currently I am trying to implement a prototype for the following problem.
I have data for machines, which sends me how long they have operated in seconds. Further, they have one sensor, which might ...
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Is there a valid way to demean a categorical variable?
I have read about use of centering an explanatory variable in regression analysis on the mean (through demeaning) when there is presence of multicollinearity. I have applied this for a binary ...
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exchangeability in LDA
Can someone help me understand how exchangeability works for Latent Dirichlet Allocation (LDA) and how it enables us to treat words in the test set that are not present in the training set?
I know ...
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Probability of Event if Another Occurs
An enemy that can drop loot has a 10% probability of spawning out of all enemies. Only one model type can drop loot out of 8 models (1/8). What is the probability of a specific enemy dropping loot, if ...
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What is the intution between point estimation of model parameters in statistics?
I am reading a topic about point estimation of model parameters. I understood it as follows:
We have "observed" a sample $X_1,...X_n$ where we know it's distribution, i.e. the sample is ...
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Expected Win (or Loss) of Base (Spin) Game Winning Combination and Bonus (Guess the Colour) Game
I'm trying to figure out the right way of calculating an expected win (or loss) for a single winnings combination of a slot base game spin with a bonus game (guess the colour).
Let's say a player ...
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Bayesian model for the reliability of a manufacturing process with noninformative prior
We have a manufacturing process $M$ with an unknown reliability $R \in (0,1)$, and, at the end of it, an automatic tool that sorts the good and bad products with a known efficiency of $E \in (0,1)$. ...
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Finding mean/std of matrix-vector multiplication [duplicate]
I have some vector $x$, with mean $\mu_x$ and standard deviation $\sigma_x$, coming from an unknown distribution. I’d like to perform the linear transformation $y = Wx + b$, where $W$ is a matrix and $...
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Distribution of the ratio of sample range to sample standard deviation for normal when n=3
Let $X_{1},X_{2},X_{3}$ be i.i.d samples from $N(\mu,\sigma^2)$.
Let $u$ denote
$$u=\frac{X_{(3)}-X_{(1)}}{S_{3}}\,,$$
where $X_{(i)}$ denotes the $i$th order statistic, and$$S_{3}=\sqrt{\frac{\sum_{i=...
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Binomial distribution conditional on the weigthed sum?
Suppose $\mathbf{X}$ is a vector of iid Bernoulli variables with the fixed success probability of $p$. The variance of X is $np(1-p)$.
Now, suppose, I am interested in the conditional probability of $...
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Average value of the minimum between $N$ positive random variables
I read this post How is the minimum of a set of IID random variables distributed? where I can find how to compute the density distribution of the minimum between $N$ positive random variables. If the ...
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Behavior of higher moments in the bivariate distribution
Given a bivariate joint distribution of random variable X and Y, $P(X,Y)$, consider the expectation value $E[(X-Y)^n]$ for different n values.
If one observes that while the variance becomes smaller ...
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Intuition for chain rule in Bayesian approach for prediction?
I am having trouble understanding the following steps in the Bayesian approach to predicting the probability of tossing a head H given some data D
\begin{aligned}
p(H \mid D) & =\int_0^1 p(H, w \...
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Representation of the expectation of absolute value of the difference $Y-X$
Given the representations of the mean values
$$E[Y]=\int(1-F(x))\,dx$$
and
$$E[X]=\int(1-G(x))\,dx$$ where $F$ and $G$ are the distributions of $Y$ and $X$ respectively,
can I use them to find the ...
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Is this lottery draw fair?
Context:
...
The public school system in the city where I live has high demand and low supply. You apply to some schools in order of preference and there is a scoring system to give priority to all ...
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Obtaining the chi-squared test statistic via geometry
I’m trying to informally derive the chi-squared test statistic using a combination of basic geometry and algebra. I’m successfully able to obtain a system of equations that contain Karl Pearson’s chi-...
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How to determine sample size of rated objects for scale development
I am currently looking into developing a scale to measure a latent construct. I believe the latent construct is a property of conversation sequences. The scale I am developing has 4 a priori defined ...
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Probability that two vertices are connected
I'm reading:
Clauset, A., Newman, M.E.J., Moore, C., 2004. Finding community structure in very large networks. Phys. Rev. E 70, 066111. https://doi.org/10.1103/PhysRevE.70.066111
There is written that ...
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Are $E_{Y|X}[Y]$ and $E_{Y|X}[Y|X]$ equivalent?
I am under the impression that:
$E_{Y|X}[Y] = E_{Y|X}[Y|X] = \int y f_{y|X}(y|X) dy$
And by extension:
$f_{y|X}(y) = f_{y|X}(y|X)$
Please correct me if I am wrong. Thank you! I understand that ...
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Probability of a given result with multiples of mixed dice with different number of faces
What is the formula to calculate the probability of getting 41 when I throw two 10-sided dice and four 8-sided dice? I’m looking for an algorithm for the general case of throwing multiples of two sets ...
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How to define & mathematically denote "true individual probability"?
The notion of an individual's "true probability" or "true risk" of a certain outcome $Y$ is contested [1, 2, 3]. Nevertheless, it is often discussed and useful for certain kinds of ...
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Using proportion to create a probability distribution
I toss a die five times
X = 1, 2, 3, 4, 5. Means there are five throws.
On each throw, we get a number on the die. This means we have 5 values for 5 throws.
Y = 1, 5, 4, 3, 2
On the first throw (X = 1)...
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Using Dudley Integral to estimate maximum singular value of Gaussian random matrices
On Exercise 5.14 of Wainwright, it provides a way to estimate maximum singular value of Gaussian random matrices using the one-step discretization bound and Gaussian comparison inequality as shown.
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Example of Failure of Hoeffding's Inequality for Empirical Risk Minimization
I am studying Introduction to Statistical Learning Theory by Bousquet, Boucheron and Lugosi. On pages 183 through 185 it considers the applicability of Hoeffding's Inequality to Empirical Risk ...
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Hypergeometric Simplification Intuition
For the final inequality, could someone explain the intution behind the $M-(N-K)\leq x$? I am struggling to find an intuitive argument for this relationship. Like if we have $N=20$ balls and $M=15$ ...
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Measuring the "Adversity" of a Probability Distribution? [closed]
In class today we saw the following example:
Imagine in a university there are 1000 students, and we know that the height of students has a normal distribution (with a specific mean and variance). The ...
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Can I Have Some Insight Into This Probabilistic Clustering Algorithm?
I'm going over past exam papers and there's a question on probability clusterin algorithms that I'm not really sure how to approach. It goes as follows:
A probabilistic clustering algorithm based on a ...
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Confidence intervals calculated from other confidence intervals (binomial problem)?
In a binomial experiment, I have an estimate for the probability of 3 independent events A, B & C, each with a 95% confidence interval.
(Trivial example values)
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Conditional expectation of a continuous random variable given a discrete random variable
For a problem I am working on I need to compute the conditional expectation of a continuous random variable $T$ given a discrete random variable $K$. I already derived a formula for the joint ...
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Example of product of sequence of random variables that does not converge in distribution to the product of the limits
One result of Slutsky's theorem is that when $X_{n}$ is a sequence of random variables converging to a random variable $X$ and $Y_{n}$ a sequence of random variables converging to a constant $c$ then ...
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Random variables implicit in Glivenko-Cantelli theorem
Theorem 1 here states $\lim_{n\to\infty} \sup_x |\mathbb{F}_n(x) - F(x)| = 0$ with probability $1$, where $\mathbb{F}_n$ is the empirical distribution function of the first $n$ $X$s, which are i.i.d. ...
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How to prove $\int_{\mathbb{R}} g(x) dF^n(x) = n \int_{\mathbb{R}} g(x) F^{n-1}(x) dF(x)$
Let $F$ be a distribution function, and let $g \colon \mathbb{R} \to \mathbb{R}$ be a real function.
I want to prove $\int_{\mathbb{R}} g(x) dF^n(x) = n \int_{\mathbb{R}} g(x) F^{n-1}(x) dF(x)$, ...
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How to compare relative weight of two subsets?
For one experiment, I have two data sources A and B contributing data points, and a function that takes the union of the sets of datapoints and assigns a quality weight to each point.
I'm trying to ...
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