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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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A question about Weight of Evidence

I have been reading about the work at Bletchley Park to crack Enigma during World War II. Part of what Turing and his colleagues did was compare pairs of encrypted messages to decide if the Germans ...
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Probability of sum of sequences of integers

Let K be a positive integer.Suppose that the integers 1,2,3,...,3k+1are written down in random order.What is the probability that at no time during this process, the sum of the integers that have been ...
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28 views

How do I prove: the posterier is proportional to the product of the prior distribution and the likelihood function? [duplicate]

In the book of pattern recognition and machine learning equation (1.66) says: ...
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Uniform distribution on $l_1$ ball of $R^n$ is not Sub-Gaussian when n is a variable?

I am self study the high dimensional probability by Roman Vershynin I was asked to show that : Uniform distribution on $l_1$ ball of $R^n$ is not Sub-Gaussian, on page 55. Exercise 3.4.9 Here is ...
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Proof for copula determined by correlation matrix

How can I proof that the copula of an elliptical distribution $El(\mu, \sigma^2, g_n)$ is fully determined by the generator function $g_n$ and the correlation matrix extracted from $\sigma^2$.
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Doubt about state of Predictor Variable in PRF and its implications [Regression Question Series - Part 1]

Given a population $(X,Y)$ we hypothesize underlying population hasa regression line as follows. The conditional expectation is $$\begin{aligned} & E(Y|x) = \beta_0 + \beta_1x & \...
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1answer
32 views

How to find the probability of a defective bulb?

Two boxes contain 20 light bulbs each. The material used for the bulbs in one of the two boxes was faulty so that one out of four bulbs go off as soon as you use them. The other box doesn't contain ...
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1answer
85 views

The likelihood function: Why is it no pdf? [duplicate]

I know that there have already been a lot of questions about why the likelihood is no probability density function and I ve read most of the answers. However, to me the point is still not clear yet ...
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probability to belong to the majority class based on observed proportions

I have a population of several categories and I don't know the proportion of these categories in my population. I want to draw (with replacement) in my population until I can say with 99% certainty ...
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1answer
58 views

Conditional Probability and its correctness

I am watching this YouTube video on conditional probability example by a university professor. He gives 2 examples: What is the probability of $2$ children being girls if we were told at least one of ...
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1answer
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How to find joint distribution of $\min(U_0,U_1)$ and $\min(U_1,U_2)$ where $(U_0,U_1,U_2)$ are i.i.d Uniform?

I have this homework question where there are 3 random variables $(U_0,U_1,U_2)$ which are independent and uniform in the interval $[-1,1]$. I have two other random variables $(X,Y)$ defined as ...
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Difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs?

What is the difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs? Example: Picking a student from class and noting his/her height and weight is several RVs (...
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Base Rate fallacy in Conditional Probability P(A|B) vs P(B|A)

From Wikipedia: P(A|B) (the conditional probability of A given B) is not equal to P(B|A). For example, if a person has dengue they might have a 90% chance of testing positive for dengue. In ...
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Is there an analytical way of determining the probability for a specific outcome given a joint p.m.f.?

I'm looking for a way to determine the probability for a specific outcome based on (what I think should probably be) a joint probability mass function. I'll try to put into words my specific case: I ...
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1answer
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Proving $X_n \rightarrow X$ in distribution implies $a+bX_n \rightarrow a+bX$ in distribution by definition

As stated in the title, I am trying to prove that if $X_n \Rightarrow X$ in distribution, then $a+bX_n \Rightarrow a+bX$ ( where $a,b\in\mathbb{R}$) in distribution using the definition as follows: $...
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Probabilistic linear regression for multiple dimensions

I have a hard time grabbing how the posterior, prior and likelihood are related in probabilistic regression. I haven't really grasped the concept. Normally I would write: $$p(f,x,t) = p(f\vert x,t)p(...
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1answer
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CDF and random variable

Please. I am trying to understand the proof, that cdf of minimum of $n$ random variables is $1-[1-F(x)]^n$ If I have $n$ independent random variables $X_1, \dots, X_n$, all of them have the same CDF $...
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Statistical distribution of randomly-selected distances

As part of a complicated project relating to widgets being sent out to incorrect factories ("missorts"), I am examining a process which involves operators standing on a conveyor belt, removing widgets ...
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1answer
28 views

Number of throws likely to have occurred the most times to get a 6 [closed]

This question has me stumped and I've been trying to figure out how to answer it for a while now - was hoping someone here could share some insight! 'The probability of getting a 6 on any throw of a ...
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1answer
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Pearl, Causality: what are variables and functional relationship?

In Pearl "Causality: Models...", he defines Causal Structure in (2.2.1) in terms of "variables" and "functional relationships". This language conflicts with standard mathematical language where a ...
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Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
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Likelihood of a linear model in matrix form

I have difficulty finding the likelihood of the data represented in the matrix form. The mapping between target variable $\mathbf{T}$ and observed variable $\mathbf{X}$ is given as $f:\mathbf{X}\...
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Joint probability distribution of correlated data points

I have a query with respect to joint distributions. Here, each output data point in $\mathbf{y}$ is conditionally independent given the inputs $\mathbf{x}$ and the mapping $f:\mathbf{x}\rightarrow \...
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1answer
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I think there is a little mistake in this exercise about the memoryless of Geometric Distribution

An exercise of Jacod and Protter: Let $X$ be Geometric. Show that for $i, j > 0$, $$P(X > i + j | X > i) = P(X > j)$$ I did it and I got a different asnwer: $$P(X > i + j | X > i) = ...
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Probability theory and counting [closed]

I am not sure how to do the following question. Please could you explain from beginning to end. Thank you. A regular insurance claimant is trying to hide 3 fraudulent claims among 7 genuine claims. ...
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Joint mass function of two perfectly correlated categorical variables

Is it possible to derive the joint probability mass function of two discrete random variables (categorical in my specific case) IF we know that they are perfectly correlated? Let's assume for ...
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1answer
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how to find correlation coefficient when X and Y follows Poisson Distribution?

A bridge is examined for corrosion. It is believed that the corrosion on left side exist is poisson distributed with mean 3 and corrosion on right side is poisson distributed with mean 1.5+0.5X where ...
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Proof of Convergence in Distribution with unbounded moment

I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much! Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
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1answer
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Exponential family form for product of Gumbel distributions

Consider the Gumbel distributions $(P_\vartheta)_{\vartheta\in\theta}=(G(\beta,\mu))_{(\beta,\mu)\in(0,\infty)\times\mathbb{R}}$ with distribution functions $$F_{\beta,\mu}(x)=e^{-e^{-\frac{1}{\beta}(...
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1answer
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Hypothesis testing when you have the entire population?

I have an experiment that involves testing the route-finding ability of 3 different critters. They have to travel between 5 different points (essentially a travelling salesman problem) and for each ...
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Number of elements in a sample space

I am looking at the number of elements in the sample space and would like to confirm something please. Using the fundamental counting principle, would it be right to say that the number of elements ...
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1answer
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What is the probability/likelihood of a sample being drawn from a probability distribution over binary values

Suppose we have a known discrete probability distribution $X$ over $\{0,1\}^k$. Given a sequence of binary values $e = (e_1, ..., e_n)\text{, where } e_i\in \{0,1\}^k$, what is the probability (or the ...
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Is there any better alternative to Linear Probability Model?

I read here, here, here, and elsewhere that linear probability model (LPM) might be used to get risk differences when the outcome variable is binomial. LPM has some advantages such as ease of ...
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1answer
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How to calculate divergence from standard deviation

I have 20 people I wish to give performance score every 6 months. I want to identify the weakest performers in the bottom 10 and remove them from the group but only those from the point of the most ...
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2answers
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Probability of the sum of cards?

The problem is basically as follows There are four cards with points [1,2,3,4], every time I draw one card randomly and then put it back to the deck. As I keep ...
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1answer
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Cross checking AI Selections vs Random Chance

Fifty Three Images AI Selects 20 images based on various criteria. I Select 20 images based on personal criteria. In a two group selection of all 53 items what is the likely overlap due to random ...
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Beginner in R programming [closed]

I face a problem when dealing with the question on the image and the value given was $n=7$, $m=11$, $t=18.72$. The detail of the question is on the picture attached. Thanks a lot
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1answer
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Distribution of number of heads by coin flipped a geometric number of times

I have random variables $N \sim \text{geo}(p)$, and $B | N \sim \text{bin}(N, q)$. I'm looking for the distribution of $B$. To be clear, I have $$ \mathbb{P}(N=n) = p(1-p)^n \quad \text{for } n = 0, ...
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Chung type LIL, integral of Brownian motion

Suppose I have two Wiener processes, which are independent - call them $B(t)$ and $W(t)$. I think it should be true that $$\liminf_{T \rightarrow \infty} \frac{\ln\ln T}{T^2}\left|\sup_{0 \leq x \leq ...
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1answer
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pdf of combined models

In a question, I have 5 systems. At a given time x, the probability that they're working is based on an exponential distribution. The combination of all systems will work if any single system is ...
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2answers
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Expected value of $g(x, y)$ without having their joint distribution

For two random variables: $X$, $Y$. If their marginal distributions are given $f_X(x)$, $f_Y(y)$ and $g(x, y)$ is some function of $X$ and $Y$. Can I get the expected value of $g(x, y)$ if I know ...
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1answer
27 views

Joint probability function of discrete variable (combinatorics)

There's a box with three types of objects: A, B, and C There are 6 of A, 8 of B, and 10 of C. At random, we remove four objects from the box I'm trying to find the joint probability $P(x, y)$ where $...
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1answer
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Why is it important that estimators are unbiased and consistent?

I am clear on the definition of unbiasedness and consistency. But why are these the criteria we use to judge whether an estimator is a good one? There are other criteria, of course, like the variance ...
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1answer
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How to find the marginal densities of the given functions

The fraction $X$ of male runners and the fraction $Y$ of female runners who compete in marathon races are described by the joint density function$$f(x,y) = \begin{cases} 8xy & 0 \le x \le y \le1 ...
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2answers
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A book on solved Probability problems

I was looking for a book to practice Probability problems with solutions to get intuition of Probability and I see Schaum's series is kind of most respected on solved problems. But I wonder which on ...
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Prove the relation between two distribution functions

I have been given a homework in a subject called "Non-Parametric Statistics" and I'm a bit stuck with it. I would be very thankful if you could give me any advice or help, which would lead to a ...
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2answers
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Good vs Bad Trial in Probability

I am watching this video on Probability, around the end there is a problem to be solved: I can't understand how a summation which extends from 1 to infinity can be equal to $\frac{1}{1-4/6}$
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1answer
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Why does the summation property of Gaussian distributions hold?

According to the summation property of Gaussian distributions: If there are 2 Gaussian distributions as given by: $y$~$N (\mu, \Sigma)$ and $y'$~$N (\mu', \Sigma')$ then $y~+y'$~$N(\mu+ \mu', \Sigma+\...
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1answer
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Drawing Gloves from a drawer — Finding Probability

Probability of drawing white glove on first attempt P(B1) = 1/5. For finding white glove at 2nd attempt means fist one was not white and hence was discarded and now 4 are left. Therefore probability ...
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1answer
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Finding number of seating arrangements

This is the question I got in Purdue University's Probability course available on YouTube: I don't understand they wrote 24 possible outcomes . Outcomes are just 5: 5 people in 5 seats. ...