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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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Bus arrival times and exponential distribution

I came across a question that is supposed to show us how the properties of the exponential distribution can be used. I know and have shown that $$P(X_i<min\{ X_1,\dots,X_n\})=\dfrac{\lambda_i}{\...
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Efficient simulation for distribution of time until coin toss pattern

Suppose we flip a biased coin (with probability $p$ being Heads) repeatedly until a certain pattern (e.g., HHHTT) appears. We are interested in the number of flips $N$ required. It is well-known that $...
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1answer
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Determine expected value for continuous random variable [duplicate]

I know that $E[X^n]$ is found by $$\displaystyle\int_{0}^\infty{x^nf_x(x)dx}$$ I simplified this to $$\displaystyle\int_{0}^\infty{ \frac{x^{\frac{v}{2}-1+n}e^{\frac{-x}{2}}}{\displaystyle\int_{0}^\...
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How is 'updating priors' in Bayesian stats different from adding more measurements to the distribution in frequentist stats?

I'm an experimental physicist so please pardon me if my thinking about this is too concrete. Let's say I am taking a measurement over and over and trying to determine the "real" value of something, ...
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1answer
41 views

expected value of money for coin toss

I play the following game using a coin that lands heads with probability $p$. I start with $X_0$ = $1$ and at each stage I gamble all I have on the toss of the coin. If it lands heads I end up with ...
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2answers
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Scoring a model with “distance from truth”

Scoring classification model performance often seems somewhat abstract (looking at you AUC scores...). There's always accuracy score, which has the advantage of being nice and easy to comprehend and ...
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1answer
30 views

If I make N trials, each independent with p chance of success, what is the probability that X or more of them will be successful?

In my problem, $p$, $X$, and $N$ are fixed and I want to know the probability. Currently, I am calculating this recursively, as for any given $p$, $X$, $N$, the probability is: ...
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Calculate the probability of an event using prior statistical information

i have a probability problem and i dont know if im using the right way to find a solution to it. Here the problem : A person drink in average 490ml of water from a bottle of 500ml and i have a ...
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1answer
40 views

CDF Variable Transformation

Let $X$ be uniform on $(-1, 2)$ and let $Y = X^2$. Find the pdf of $Y$. So far I have noted that $F_X(x) = P(X \leq x) = \int_{-1}^x \frac{1}{3} dt = \frac{1}{3}(x+1)$. Then, since $Y=X^2$, $y \in [...
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Dependency of a trait A on trait B

Given a species of animals. There are two groups of animals. One exhibiting trait A and one not exhibiting trait A. A certain percentage of animals with trait A also have trait B and a much smaller ...
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1answer
30 views

Estimating probabilities from truncated set of counts?

If I have a truncated histogram like this, where the ordinate is the number of occurrences $N_X$ of each value of $X$, and the abscissa is the value of $X$, is there a way to estimate the probability $...
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1answer
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Calculate the probability of any one independent scenario occurring

I have $3$ independent scenarios and I need to calculate the probability of any one of them occurring. I’ve calculated the probability of the $i$-th event as: : $1-\big(\frac{70000-1}{70000}\big)^1 ...
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When is the pmf of the difference of two independent random variables symmetric in zero?

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
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0answers
50 views

probability measure of probability measure

I if denote with $\Delta(X)$ the set of all probability measures over $X$, what is the meaning (if there is any) of $\Delta(\Delta(X))$ ?
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1answer
24 views

Binomial distribution for randomly drawn probabilities

Setting Probability theory can be a weird place sometimes. Here I was, confident in my insane math skills, trying to solve the following problem: Let $N, \alpha$ and $\beta$ be given. ...
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Meaning of probability as used in machine learning

In machine learning, it seems that people are often happy to call the output of any function (e.g. logistic function) with the range $[0, 1]$, a probability. How correct is that thinking? How ...
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2answers
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What can we say about $N_{i}$ where $N=N_{1}+\cdots+N_{m}$, $N\thicksim Geom(\frac{1-p}{p})$ and conditional distribution of $N_{j}$ is binomial

Suppose that the number of events $N$ is a Geometric random variable with mean $\frac{1-p}{p}$. Further suppose that each event can be classified into one of $m$ types with probabilities $p_{1},p_{2},\...
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Time-series forecasting of weakly correlated, univariate series

Given an event that happens with a probability of $\lambda_1$, and another event happens with a probability of $\lambda_2$, what is the probability that they both occur? I have a dataset of ...
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0answers
20 views

Why H(y) is a concave function of p(x) when p(y) is a linear function of p(x)?

I got stuck in the line when reading a theorem in The Elements of Information Theory on page 59: If p(y|x) is fixed, then p(y) is a linear function of p(x). Hence H(Y), which is a concave ...
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In a bootstrap sample, what is the probability that the 5th observation is the same as the 1st observation?

I know that the probability of jth observation being in a sample is: p(jth obs in bootstrap sample) = (1 - 1/n) ^ n p(5) = (1 - 1/n)^n p(3) = (1 - 1/n)^n Can you give me a hint on how to go about ...
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Statistics and Probability. Please show solution [closed]

Miss Romero noted that the mean scores of a random sample of 15 grade 8 students who had taken a special test were 80.5. If the standard deviation of the scores was 3.1 and the sample came from an ...
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1answer
67 views

What is expected times of sampling that a full population is covered?

There is a population with N instances. A sampling means we draw randomly M instances from the population without replacement. ...
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59 views

Win probability of a game character vs. two other characters

Clarification about original problem: Dota 2 is played in matches between two teams of five players—known as the Radiant and Dire—, with each team occupying and defending their own separate base ...
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1answer
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Average time for a random walk on the edges of a cube

In a interview i had something similar to this question Random walk on the edges of a cube. But this time there is only a ant, it takes one minute to go through a single edge. The ant can use the ...
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2answers
60 views

can one calculate a p value in a distribution [closed]

I have an event which happens so many times, I want to know how significantly is 1300$ in comparison to the distribution. which test should I use ?
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variational lower bound confusion

In this blog describing variational inference under the section KL divergence and ELBO they mention that in the equation $$p(x) = \frac{w(x)}{Z}$$ we can substitute $w$ and $Z$ with: $$Z = p(x;\...
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8answers
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Do error bars on probabilities have any meaning?

People often say some event has a 50-60% chance of happening. Sometimes I will even see people give explicit error bars on probability assignments. Do these statements have any meaning or are they ...
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0answers
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compare Bayesian linear regression vs standard linear regression

1st question, I recently learnt bayesian linear regression, but I'm confused that in what situation we should use bayesian linear regression, and when to use standard linear regression? What is the ...
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1answer
53 views

A random variable with a step function CDF is discrete?

Consider the step function $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$ $\mu_j\in \mathbb{R}$ $\...
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0answers
27 views

Confidence intervals on multiplication of two means with confidence intervals

I have survival probability 0.5 with a confidence interval around it. I have mean number of offspring 15.1 produced with confidence interval around it. I want to find species "fitness" by ...
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1answer
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Is it possible to compare probabilities of 2 logistic different models with same data set? [closed]

I have two logistic regression models (A&B), one gives probability of "person A" buying "Product X" and other model gives probability of person A buying "product Y". The training data for both ...
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How to derive the maximum *a posteriori* estimate when the prior distribution is Normal $N(m,r^2)$?

I am learning probabilities and I need a guru to help with this problem: Assume $p(y | x) = N(ax,\ s^2)$, where all quantities are scalars, $a$ and $s$ are known constants, and the prior ...
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1answer
33 views

Expected value using indicator variable

Suppose that $8$ white balls and $2$ black balls will be randomly ordered, from left to right (with all permutations of the $10$ balls equally likely), what is the expected value of the number of ...
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Conditioning the probability obtained from a machine learning model

I have developed a random forest classifier to predict whether a customer will churn. The data used to produce this model has the following form ...
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1answer
118 views

Do Stochastic Processes such as the Gaussian Process/Dirichlet Process have densities? If not, how can Bayes rule be applied to them?

The Dirichlet Pocess and Gaussian Process are often referred to as "distributions over functions" or "distributions over distributions". In that case, can I meaningfully talk about the density of a ...
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0answers
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Conditional distribution with Jeffreys’ Prior [closed]

If $\pi(\mu,\sigma)$ corresponds to $N(\mu,\sigma^2)\times\mu^{-1/2}$, what is $\pi(\mu|\sigma)$?
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1answer
39 views

Poisson question that I can’t solve [closed]

1.Customers arrive at a bank at a poisson rate. Suppose that two customers arrived during the first hour. What is the probability that (A) both arrived during the first $20$ minutes? (B) at least ...
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1answer
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SOA Conditional probability question [closed]

I can't get this correct. Could you please provide a hint to aproach this problem? How do i get the probability of that loss? "A state is starting a lottery game. To enter this lottery, a player ...
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1answer
21 views

consistency of an estimator in real data

as we know in statistics we are interested in the properties of an estimator , as my estimator is consistent , in probability sense it can be shown as $P(|estimator - true| <- positive) = 1$ as ...
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1answer
27 views

Using a generalised linear model for tennis winning probability

I have a dataset of historical tennis matches, one row per match. Each row has the ELO points of each player, and a calculated ELO difference. And of course a column to indicate if Player 1 won as a 1 ...
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1answer
39 views

Combining probabilities to find most probable window

I have a series of observations, with an associated probability that an event is occuring at timestep t, something like: [0.8, 0.8, 0.3, 0.9, 0.2] Events can ...
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1answer
47 views

Multinomial logistic regression: How to calculate the baseline probability that increases by Exp(B)?

I have troubles finding an answer to my question, regarding a multinomial logistic regression model, with a single continuous predictor. Let's say these are the absolute frequencies in the ...
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0answers
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Book “introduction to probability 2nd edition” example 1.25 “Grade of Service”, I can't get same result

I am in self-study probability theory. I wrote a python script to calculate $p(15)$ in example 1.25 and got a result '0.05012925454322169' which is '0.0399' in this book. Here is the example: An ...
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0answers
33 views

binomial pairs probability

A group of four people is said to be “interesting", if there are at most five pairs who are friends. Assume that each pair of people are friends, independent of every other pair, with probability 1/2 ....
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KL divergence between sample and true (multivariate normal) distribution

I was wondering, whether there is a possible interpretation of the KL-Divergence between sample and true distribution in terms of probabilities. E.g. given $P=\mathcal{N}\left(\mu,\Sigma\right)$ and $...
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Calculate number of points with expected values

The Problem Lets say you have a game where using an item gives you a certain number of points, with different probabilities for each point value. The probabilities are unknown, but can be estimated ...
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4answers
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Question regarding independence of events

I am reading Wasserman's book and his class notes. In the notes, I found the following statement If A, B are disjoint, both having positive probability, then A and B cannot be independent. Now, if ...
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0answers
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at what probability will the probability we start considering the data?

For example I have this problem, Do Americans tend to vote for the taller of the two candidates in a presidential election? In 30 presidential elections since 1856, 18 of the winners were taller than ...
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0answers
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Finding Probability of a digit given a sequences?

I have a n sequences of numbers ranges from 1 to 4, say sequence s1 = [1,3, 1, 4] and s2 = [2,1,3,4] up to s(n). My question is how can i find probability of a number coming right after a sequence, ...
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1answer
40 views

joint PMF in a class of n students

A class of n students takes a test in which each student gets an A with probability p, a B with probability q, and a grade below B with probability 1 − p − q, independently of any other student. If X ...