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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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Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
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1answer
38 views

Mutual Independence in a Multivariate Normal with Identity Covariance

Consider a random vector $X$ which follows a multivariate nomal with zero means and Identity Covariance. $X\sim \mathcal{N}_n(\mathbf 0, \mathbf I)$ We can say that the individual variables $X_1, ...
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2answers
64 views

Distribution of maximum of normally distributed random variables

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far: $$ \begin{align*} F_Y(y) &= \mathbb{P}(\max(...
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24 views

CLT and convergence of Variance

I am looking at a problem where the sum of the individual $X_i$ is $S_n=X_1+\dotsm+X_n$. The probability is given as, $P(X_i=i)=P(X_i=-i)=\frac{i^{-\alpha}}{4}$ and $P(X_i=0)=1-\frac{i^{-\alpha}}{2}$. ...
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7 views

coverage index?

Suppose I have a space of potential outcomes X with a probability distribution on it. I assume that there is a distance function between elements of X (e.g. X is a metric space). I also have a set S ...
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11 views

Statistical Application

wasn't sure where to ask this question but figured StackExchange would be helpful. I was watching an NBA basketball game that made me think about statistics: if the Lakers (15-10) and Grizzlies (15-9) ...
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I have this as an assignment question and can not figure it out, need some help please [on hold]

a)The mean weight of King Penguins found in an Antarctic colony last year was 15.4 kg. In a sample of 35 penguins same time this year in the same colony, the mean penguin weight is 14.6 kg. Assume the ...
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18 views

Expectation of Gaussian random vectors [on hold]

Given a random vector $x\in \Bbb R^{n \times 1}$ with $x\sim N(\mu, I\sigma^2)$, what is the expectation of the following vector: $ E\left(\frac{x}{\| x\|_2}\right)\;?$ It is known that when $\mu = 0$...
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1answer
49 views

Basic probability theory

I was recently given the following statistics: On a particular highway, 18% of drivers are black, 63% of drivers searched by the police are black. So, a black driver is 7.7 times more likely to be ...
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Shouldn't the joint probability of 2 independent events be equal to zero?

If the joint probability is the intersection of 2 events, then shouldn't the joint probability of 2 independent events be zero since they don't intersect at all? I'm confused.
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General techniques for coupling a set of random variables with mutual dependence

Disclaimer: the usage of coupling is in the title is not of the usual definition in probability theory. Suppose I have a set of random variables $\{X_1, X_2, \dots, X_n\}$, indexed by time $t$, and ...
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Latent class clustering in R

I apologize if a similar question has already beeen asked. I am trying to do some cluster analysis using both categorical and numerical variables. There are ways to do so by using k-prototypes or ...
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0answers
23 views

Rewrite a constraint on the probability distribution using the cumulative distribution function

Consider a probability distribution $P: \mathbb{R}^3\rightarrow [0,1]$ and assume $$ (\diamond) \text{ }\int_{(x_1,x_2,x_3)\in \mathbb{R}^3 \text{ s.t. } x_3= x_1-x_2} dP=1 $$ Questions: Is there a ...
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1answer
23 views

Binary classification and target-label proportion

Suppose that we have a binary classification problem with a vector y = [1 1 0 1 0 0 1 ... 0] having the proportion: ...
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Who invented the independence notation $\perp \!\!\! \perp$?

This is more of a historical question: who invented the notation $\perp \!\!\! \perp$ for denoting (conditional) independence?
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Main differences (problems and mathematics) between traditional statistics and high dimensional statistics

High dimensional statistics seems to be hot nowadays. What are the main differences, in terms of questions and problems it tries to solve, as well as the mathematical tools used, between "traditional"...
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Statistics Question regarding probability distribution function [on hold]

Suppose that two random variables, X and Y have a continuous joint distribution with the following joint probability density function (pdf): 𝑓(𝑥, 𝑦) = 3/8*(2𝑥 + 5𝑦^2), zero < 𝑥 < 1; zero &...
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1answer
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How to generate random variables which are correlated and yet marginally identically distributed? [on hold]

I am wondering if there is a process to which we can generate random variables where there is a correlation structure between them, yet they are still marginally identically distributed? One idea that ...
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0answers
16 views

Estimating true frequency from a finite sample

Say I have a process that follows a multinomial distribution such that probability of observing outcome $i$ is $p_i$. However I don't know how many $i$ there are, and I don't know the $p_i$'s. I want ...
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2answers
42 views

Probability that a sample came from a known distribution

I'm looking for a general solution to what I assume must be a common problem because it comes up in every Bayesian calculation, but doesn't seem to be directly answered anywhere. I have an extremely ...
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1answer
52 views

Poisson distribution - Number of accidents

I need help with this probability problem. The number of fatal car accidents that happen in a specific region follows the Poisson distribution with a rate of 0.5 fatal car accidents per day ...
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1answer
40 views

How to derive joint CDF Gumbel distribution

If you have 3 random variables: $X$, $Y$, and $Z$ and they have independent Gumbel distribution. $A$, $B$ and $C$ are three discrete random variables that are functions of $X$, $Y$, and $Z$ as per the ...
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0answers
31 views

Is there a distribution such that a sum of its samples creates the uniform distribution? [duplicate]

Say I sample $N$ numbers from a mystery discrete distribution and add them all together. My goal is for the sum of these random variables to be uniformly distributed from $[a,b]$ and thus the ...
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2answers
37 views

How to swap variables in a conditional normal distribution?

I assume that I have two normal distributed variables where one depends on the other: $P(A) \sim N(0,\sigma_a)$ $P(B|A) \sim N(q\cdot A, \sigma_b)$ How can I get the reverse $P(A|B)$ assuming that ...
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Correcting biased probability from sample of only cases with at least one occurence

I have purchasing data over time for individuals, and am trying to figure out the annual probability of people in a given age cohort making a purchase. (Max purchases/year is 1). The problem is that ...
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Question about necessary experimental recruitment size given particular design

Suppose each individual faces a series of test-rounds, each a part of a broader task. In each round, they face some probability that the task ends, and some probability they continue to the next round....
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1answer
77 views

Proving a remainder term converges to 0 in probability

So we have these definitions: σ̂^2_1= (1/n)∑(Xi−μ)^2 σ̂^2_2= (1/n)∑(Xi−Xbar)^2 I have shown that n^0.5(σ̂^2_2−σ^2)= n^0.5(σ̂^2_1−σ^2)- n^0.5(Xbar-μ)^2 I am trying to show that the remainder term ...
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0answers
20 views

Toss coin simulation in R [closed]

I need help doing this in R. If we have an unfair coin is tossed 100 times for a total of 40 heads. Using R, I need to simulate a 95% confidence interval for p, the probability of obtaining a head for ...
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0answers
12 views

Variance reduction [closed]

I need to calculate the value of double integration using R: - Using the raw estimator - Using antithetic variables - Using control variates - Compare the estimates. Which is best?
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1answer
48 views

Understanding the shifted log-normal distribution

I have difficulties understanding why a third parameter (the shift) is necessary to describe the log-normal distribution. Let's say we have a normal random variable X, if I shift this variable by an ...
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1answer
19 views

Marginal from conditional given no parameter reliance

If $Y|X \sim \text{Normal}(0,1)$ is it true that $Y \sim \text{Normal}(0,1)$. This intuitively seems true as the Normal is characterized by the mean and variance, which have no reliance on $X$. So no ...
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1answer
56 views

Convergence of $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$ - central limit theorem

Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$, where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\...
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0answers
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Proving the characteristic function involving sequence of 2 iid variables [duplicate]

We have an iid sequence with: $X_1,X_2,...$ with mean $\mu$ and variance $\sigma^2$. We have another similar sequence: $$Y_1,Y_2,...$$. We have a sequence $U1,U_2,...$ where, $U_n=1/\sqrt(2(n\sigma^...
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Bayesian inferences on related conditional uniforms [closed]

I'm having issues with conditional probabilities that are uniform in Bayesian inference. Assuming there to be a R.V $X\sim \operatorname{Uni}(0,A)$ where $P(X) = 1/A$ when $ 1\leq X\leq A$. 0 ...
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0answers
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A PROBLEM IN SAMPLING THEORY [closed]

Given the sample design with probabilities: P((1,2)) = 0,1; P((2,1)) = 0,2; P((1,2,3)) = 0,4; P((1,2,3,4)) = 0,1; P((3,2,4,1)) = 0,2 and the following linear estimator of the total T (Y) e(1,2) =...
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2answers
38 views

Prove convergence in distribution for n times the minimum of an unknown positive distribution

Let $Z_1, Z_2, ...$ be independent and identically distributed random variables with some density $f$. Suppose that $P(Z_i > 0) = 1$, and that $$ \lambda = \lim_{x\to 0} f(x) > 0$$ Let $X_n = ...
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0answers
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Prove bi-directional relationship between convergence in distribution and convergence of probability mass functions

Let $X$ be a random variable that is positive and integer-valued. Let $X_1, X_2, ...$ also be random variables that are positive and integer-valued. Prove that $X_n$ converges in distribution to $X$ ...
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1answer
31 views

Optimization using the optim function in R with a two parameter exponential distribution

I'm having trouble trying to optimize a two-parameter exponential distribution, by finding the maximum likelihood function and then using the function optim() in R ...
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1answer
25 views

Can we conclude, with the strong law of large numbers, that $n$ random variables are independent? [closed]

Suppose we have a sequence of identically distributed random variables $X_1, \ldots, X_n$, and that we know $(X_1 + \ldots + X_n)/n$ converges almost surely to $\mu = E[X]$ as $n$ approaches infinity. ...
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1answer
24 views

Confused about type of analysis between categorical variable (degree of severity of accident) and continuous variable (number of car passengers)

Good afternoon, As indicated in the title, for a project I am interested in finding any (if at all) relation between the number of passengers in a car and the severity of an accident. I am using JMP ...
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0answers
18 views

Finding the score for an event of interest

I would like some inputs in a problem I am trying to solve, which I don't know if I should go all the way to supervised learning or general statistics: I have data behavior data about how customers ...
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0answers
11 views

Pros and Cons of Very Sparse Random Projection when s is large [closed]

I have question about VSRP. Based on sparse random projection with s = 3 from wiki. With very sparse random projection, s >> 3 r(ji) = square_root(s) {1 with prob 1/2s; 0 with prob 1 - 1/s; -1 with ...
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1answer
51 views

If you roll eight 12-sided dice at one time, what is the chance you get five of any one number?

If you roll eight 12-sided dice at one time, what is the chance that you get five of ANY one number? What is the chance that you get five of a SPECIFIC number, say, number 2?
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1answer
21 views

Gradient-Descent for Parameter search

I have a rather straight-forward algorithm for finding the maximum-likelihood parameter of a probability distribution using sub-sampling. I'm fairly confident this algorithm is not novel and was ...
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0answers
19 views

Is it better to have 1 washer room in an apartment with 20 washers or 10 floors with 2 washers each in a dorm?

The housing department of the university I study at gave a presentation on a new dorm being constructed slated for 2022. Being in the preliminary phases, they are currently designing how this building ...
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0answers
53 views
+50

From trivariate cdf to the distribution of differences of random variables

Consider a trivariate cumulative distribution function (cdf) $G$. Is there a collection of necessary conditions on $G$ ensuring that $$ \exists \text{ a random vector $(X_1,X_2)$ such that $(X_1, ...
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1answer
20 views

calculation of paramters needed for joint probability distribution?

Please correct me if I'm wrong. From my understanding, the number of entries in the above image is 7 because you need to calculate 7 and the 8th one can be done by 1-p. But I can't understand how ...
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0answers
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How to measure the “variability” across a set of many (>>2) probability distributions?

Given a set of many of discrete probability distributions, is there a way I can efficiently calculate a metric that quantifies how different the entire set of these probability distributions are to ...
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0answers
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Can you assign a 50% probability to an unknown situation?

Say I have a bag of 5 balls. I know there is some mixture of red and black balls in the bag. I pull out 4 balls and replace as follows: Red, Black, Red, Black. Can I assume there is a 50% chance the ...
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1answer
66 views

Bayes theorem: a function of D, or theta, or both?

In the Bayes formula as written for machine learning applications, $$ p(\theta|D) = \frac{ p(D|\theta) p(\theta) }{ p(D) } $$ where $D$ is the data, $\theta$ are the model parameters. Commonly $p(\...