Questions tagged [probability]

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

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Marginal distribution of a Poisson rate

I have a random sample of Yi~Poisson(θ) with i=1,..,n. Where I assume a gamma(a,b) prior for θ. Where I'm able to find the marginal distribution of Y to be a NegativeBionomial(a,p) where p=(1/(1+b), ...
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How the variance of a potential loss X has been derived

I'm studying Insurance and I have a question about how the variance has been computed in this example. Imagine a case where an "agent" may suffer a loss, because of an event (an accident) occurring ...
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19 views

Basic probability question in microeconomics [closed]

There are two players in a gamer N={1,2}, their actions lay between a=[1,infty) and their type lays between [0,200]. given that x is within [0,200] and P(t_2 < x|t_1)=x/200 (the type of each player ...
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Factorization of the Characteristic function for sum of dependent data [closed]

We know that for $S_n=X_1+X_2+...+X_n$ the characteristic function is $\mathbb{E}_X\left\{\exp(iuS_n)\right\}$, where if $X_1 ,X_2 ,..., X_n$ are independent, it can be expressed as follows \begin{...
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1answer
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Understanding three prisoners (Statistical Inference - Cassella and Berger)

There are quite a few questions regarding three prisoners riddle, but my question relating to a particular line of reasoning from Cassella and Berger, as highlighted in the image below. Could you ...
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Extendability of finite sequences of exchangeable random variables?

The old paper Binomial Mixtures and Finite Exchangeability by G. R. Wood makes reference (see Table 2) to a 1971 conjecture of Crisma that gives a formula for the probability that an exchangeable ...
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Help answer these questions [closed]

Can you please answer these questions
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6answers
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Expectation of 500 coin flips after 500 realizations

I was hoping someone could provide clarity surrounding the following scenario. You are asked "What is the expected number of observed heads and tails if you flip a fair coin 1000 times". Knowing that ...
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Combinatorics and probability

I have 7 interviewees and 14 interviewers. Each interviewee is interviewed by 4 of those 14 interviewees. Is it possible to have every interviewee to be shared by at least 1 interviewer, as to allow ...
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35 views

Percents of Random Samples

I have three sets $A,B,C$ of sizes $N_A=2508$, $N_B=36211$ and $N_C=2296$ respectively, containing binary values. I took 200 samples of each set to produce point estimates of the averages: $\hat p_A=0....
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Which come first? (random walks)

Suppose I have a continuous time random walk in one (non-time) dimension, based on not-necessarily-Gaussian white noise. I know its value at the beginning and end of an interval, and from inside of ...
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31 views

random walk with probability [closed]

I just have no idea how to do this. Can anyone give a hint? Thanks for your time!
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1answer
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Subset of the event {$i$ aces in $i$ cards} and its probability

As far as I'm aware, $A$ is a subset of $B$ if $x\in A$, then $x\in B$ for all $x\in A$. If so, how could this statement (please see the image below) say the event {4 aces in 4 cards} was a subset of ...
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Distribution of X/(X+Y) when X and Y are independent, but non-identical Binomial distributions [closed]

$X \sim Bin(n, p)$, and $Y \sim Bin(m, q)$, then what is the distribution of $\frac{X}{X+Y}$?
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1answer
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Why Pareto distribution has pointy head? How its symmetrical version looks like?

I want to use Pareto distribution for the stock price probability distribution. They say (B. Mandelbrot and N. Taleb Mild vs Wild Randomness) it represents the price changes, especially the tail ...
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How to compute the probability for a firm to move from on group to another in a panel database?

I have a panel database for firms i (90 firms) across year t (from 2013 to 2019). Some firms witnessed a cross border investment in a year that is between 2013 and 2019. I created a dummy=1, when the ...
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Bayesian Question [closed]

A person belongs to 5 different groups. Group A consists has 10 people, 6 of them male. Group B consists has 20 people, 11 of them male. Group C consists has 30 people, 16 of them male. What is the ...
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Approximate posterior distribution [closed]

Let $\mathbf{a_k} \sim \mathcal{GP}(\mathbf{m}_k,\mathbf{C_k})$ where $\mathbf{a} \in \mathbb{R}^T$ is modeled as Gaussian process with mean $\mathbf{m} \in \mathbb{R}^T$ and prior covariance $\mathbf{...
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What information in general is necessary to fully specify a multivariate distribution?

Given some multivariate probability distribution, we can fully describe it with its density or mass function -- we can associate each point in the space with either a probability density or mass, ...
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1answer
64 views

Obtaining valid win probabilities from contest data using a binomial model

I conducted an enclosure experiment on lizards where I recorded contest outcome for every male-male combat. We had three morphs of lizards (o, w, y) in each enclosure. I am interested in obtaining ...
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1answer
56 views

Logit - probit regression

I was running regression of - determinants of acceptance into a social science college. I found this unrelated paper (Screenshot of relevant page attached herewith). Here, they have computed logit and ...
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Finite Mean and Finite Variance [duplicate]

Let $X$ be a random variable. Suppose there exists a constant $c ∈ R$ such that $E(|X − c|^2) < ∞$. Show that the random variable $X$ has finite mean and variance. And I'm quite confused about the ...
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One-way ANOVA method validation

Here is my case: Let's say we have 50 polygons(looks like this:and a point set distributed within these 50 polygons. So that for each polygon, there is an associated point density. What I want to ...
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1answer
28 views

How many losing runs of length r playing x numbers in roulette in a sample of n bets?

A losing run is defined as a number of consecutive losing bets. I've written a program to simulate this but the results I'm getting are different from what the formula gives, so one or another is ...
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How many times would you need to watch a movie to have seen every moment of it (i.e. not blinking)?

Say we blink stochastically for .1 seconds every 4 seconds on average. How many times would we need to watch a 120 minute movie (assuming your eyes are glued to the screen the whole time), to have a ...
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1answer
37 views

Where is the measure theoretic probability theory actually applied?

Where is measure theoretic probability theory actually applied? I've done quite a bit of graduate work in machine learning, Bayesian machine learning, information theory, and statistics (both ...
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Calculate conditional probability using sample from 3variate normal distribution

I am working in R and have a matrix (200X3) named X where I have simulated data from a 3variate normal distribution given its parameters (variance-covariance matrix & and mu). My task is to ...
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22 views

How high is the probability that copy N bytes of data from /dev/random to a file will result in a working application? [closed]

After talking with some colleagues, a question came up about the probability of copying N bytes from /dev/random to a given file on disk (either creating a new file, or truncating and overwriting an ...
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29 views

Expectation of the k-th order statistic of a standard Gaussian sample

Let $(X_1,\dots,X_n)$ be independent random variables with common distribution $\mathcal{N}(0,1)$. The order statistics satisfy $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$. I am interested in the ...
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1answer
16 views

How to rank data based on cross-validation

I had this problem from a long time. I have small dataset with about 1000 data points. The data is labeled as 1 or 0 (i.e. ...
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11 views

Estimation of generalized gamma convolutions

How can i estimate on a data sample parameters of a generalised gamma convolution ? To be more specific, if my estimation gives me only a gamma convolution and not a generalised gamma convolution i'll ...
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1answer
16 views

Range of probability for non-independent events

Suppose the probability of rain on day 1 is $p_1$ and probability of rain on day 2 is $p_2$. Then the probability of rain for the entire two-day period is $1-(1-p_1)(1-p_2)$, under the assumption that ...
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1answer
22 views

Understanding t bounds on MGF

I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook: Suppose X is a random variable for which the pdf is: ...
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33 views

How to change the distribution of classes?

I have two classes [0,1] and I want to evaluate algorithm on different distributions of classes. I did label flipping incrementally such as 0%, 10%,20%,....,90%,100%. Does label flipping change the ...
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1answer
25 views

Moment of a function of Gaussian random variables: $\mathbb{E}[(a_{i}^{\top}AA^{\top}a_{j})^{q}]$

Let $A$ be an $m\times k$ matrix with iid $\mathcal{N}(0,1)$ entries and $a_{i}$ and $a_{j}$ be its $i$th and $j$th columns. I would like to compute the following quantity: \begin{equation} \mathbb{E}...
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Probability density function analysis question

The distribution(normally distributed) of the amount of money that people withdraw from an ATM has an average of \$70 with a standard deviation of \$10. If a group of 15 people withdraws money in a ...
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Determining the degree of confidence of a Prediction

I'm handling a text multi class classification problem using RNN LSTM and model is pretty accurate (approximately 95%). But I'm interested in finding the degree of accuracy of a Prediction. Can I make ...
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1answer
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Stochastic Models (probability)- simple symmetric random walk question

My study group and I are currently stumped on this probability question related to Stochastic models. Let {Xn} be a simple symmetric random walk (ie p = 1/2). ) Give an approximation of the ...
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How do you prove that samples are equally distributed even when taken without replacement? [closed]

I was reading Casellar and Berger's book and came across this well-known property. Although, in the book, the proof for any Xi is not provided. Does anyone know how to follow up on it? I'm curious, is ...
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2answers
57 views

Probability of width of a confidence interval being less than a value [closed]

Let independent random variables $X1, X2,…, Xn$ be identically normally distributed with unknown mean $μ$ and unknown variance $σ^2$: (a) Determine the 95% confidence interval for μ, $n=51$, $\bar{x}...
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1answer
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Putnam problem - guess an integer in an odd number of attempts

I have no clue how to answer this question. This is 2002 Putnam examination question B-4. B4 An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with ...
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1answer
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Is a range of values from an exponential distribution still exponentially distributed?

I have to generate numbers of two different exponential distribution ($e_1, e_2$) with parameters respectively $\lambda_1$ and $\lambda_2 = k \lambda_1$, with $0<k<1$. But I also want to ...
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1answer
16 views

Calculate the difference between the multinomial probability

I have an example data as follows: ...
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Powerlaw Distribution and old nodes [closed]

I am working on the size of discussions (number of replies, replies to the replies) on Twitter data, and I am observing Power-law distribution for small and medium-size discussion but it is not valid ...
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1answer
49 views

Sum of all probabilities is not 1 in homework

Excerpt from "A first course in Probability, 8ed" A student is taking a one-hour-time-limit makeup examination. Suppose the probability that the student will finish the exam in less than $x$ hours ...
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1answer
25 views

Conditional Poisson Process

I cannot reach a correct answer and I don't know why. I am trying to calculate this by conditioning on $N(t)=n$ and I ended up with $e^{-At(z^s)}$. However, the correct answer is $\dfrac{e^{-At(z^s)}-...
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1answer
37 views

What prior would lead to $\ell_\infty$ regularization of model weights?

Gaussian prior on weights of a GLM lead to Ridge / $\ell_2$ squared regularization. Laplace prior leads to $\ell_1$ regularization Question What prior would lead to $\ell_\infty$ regularization ?
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How can I calculate my winning percentage?

The racehorse time follows a normal distribution. Horse A Median T1 standard deviation σ1 Horse B Median T2 standard deviation σ2 If this is the case, how do you calculate the probability that ...
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1answer
27 views

Help - Expectations and Ratios

I would need your help for a problem I have and I don't know how to solve. I would like to know whether I could prove that : $$E[0.5X/(0.5X+0.25)] = E[0.5(1-X)/(0.5(1-X)+0.25)]$$ knowing that $E[X] =...
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1answer
64 views

Ratio of two independent normal : cumulative sum [duplicate]

Given two independent normal variables, $X\sim N(0, \sigma^2_X)$ and $Y\sim N (0, \sigma_Y^2)$, find the probability that $\frac{X}{Y} < 1$. I seem to prove: $Z = \frac X Y \sim $ Cauchy ...