Questions tagged [probability]

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

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50 views

Prove that $E[e^{2(m−1)X^2}]\le m$

I'm reading "Understanding Machine Learning: From theory to algorithms". The problem is as follows, which is Exercise 31.1 of the book on page 416. Let $X$ be a random variable that satisfies $P[X \...
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1answer
46 views

Parameter estimation under Gamma noise distribution

I have a model as follows: $y_i = \theta x_i + \eta_i, i=1,2,...,N$ where $y_i$ and $x_i$ are known observations greater than $0$, the $\eta_i\sim Gamma(a,b), a>1$. Now, I want to obtain the ...
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1answer
36 views

Account for the number of Bernoulli trials in the response and selected predictors of GLM?

I have sampled animals at a number of sites and time points. Each site is sampled up to 4 times a year and is sampled over multiple years. At each site*time ...
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35 views

Does the joint pdf $f_{x, y} (x, y)$ equal to the conditional $p_{y | x} (y | x)$ for all random variables?

So I have this question where you are given two random variables, $X$ and $Y$. $X$ is a continuous random variable (represented as a mean) with a distribution of $Exp(1)$ (exponential with $\lambda = ...
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26 views

Convergence of series of dependent random variable, central limit theorem

My friend and I have a problem on central limit theorem. Given $X_1,X_2......$ are i.i.d random variables with mean $\mu$=0, variance $\sigma^2=1$(may or may not be normally distributed). If we ...
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1answer
21 views

How can I find $P(X/Y \le t), P(XY \le t), P(XY/Z \le t)$?

In preparation for my final exam, I am trying to generate problems to solve. My professor suggested that I play with independent uniform distributions. So here are three probabilities that I have not ...
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10 views

Parsing and understanding plate notation for topic modeling example?

I'm trying to understand the following plate notation which is used a lot as an example of topic model to introduce variational methods, etc. I wanted to ask if my understanding is correctly depicted ...
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16 views

Maximum entropy prior for dichotomous variables [closed]

I have a set of dichotomous variables $A, B, C,$... and I know their probabilities $P(A), P(B), P(C),$... as well es their pairwise dependencies $P(A \cap B), P(A \cap C), P(B \cap C),$... . Or in ...
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13 views

Stationary Distributions of a irreducible Markov chain

I was trying to get all Stationary Distributions of the following Markov chain. Intuitively, I would say there are two resulting from splitting op the irreducible Markov chain into two reducible ones. ...
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1answer
22 views

If Distribution is Geometric does that mean underlying probability of success for each trial p is fixed?

we know that if p (probability of success at each trial is fixed then the probability of each trial, then probability of first success at kth trial is given by Geometric Distribution I need to ...
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66 views

A simple question about expectations

@psboonstra This is a valid point. After posting it, I too realized that the question is ill-posed. I attempted to oversimplify a problem that I had encountered in finding a characteristic function of ...
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23 views

Expected wait time in two different types of banks?

Problem Statement There are two banks both have 100 people inside who want to cash their check and must do so by talking to the teller. In both banks, there are 10 tellers. In Bank 1 all 100 people ...
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13 views

Producer risk vs. consumer risk in gauge R&R

I did a gauge R&R study. In order to evaluate the data, I used the model $$y = part + operator + part:operator + \epsilon$$ where $part$ and $operator$ are random effects. Next, I used ANOVA ...
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1answer
52 views

Distribution of uniform RVs under sum constraint

Suppose I generate $x_1,x_2,x_3,x_4$ through the following procedure: Sample $x_1,x_2,x_3 \sim \text{unif}(0, 1)$, iid While $x_1+x_2+x_3 > 1$, resample them all Let $x_4 = 1 - x_1 - x_2 - x_3$ ...
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1answer
78 views

Average number of random permutations of a sequence, before seeing a sorted sequence

Suppose you have a sequence of N numbers. You check the sequence and see if it's sorted in ascending order . If not, you shuffle it (i.e., randomly permute the order of its elements) and check again. ...
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2answers
64 views

Finite $k$th moment of a function of random variable

Let $X = a/h$, where $X$, $a$ and $h$ are random variables, with $X$ an i.i.d. sequence. If $X$ has finite 8th moment, can we infer that $a$ has finite $8$th moment as well? Thanks
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Poisson Distribution vs Binomial Distribution

I have a specific problem here. It is stated as: "Assume that 30% of students in a university take public transportation daily to commute to their college. Suppose 10 of the students are randomly ...
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Lagrange Multiplier (SCORE) test verification?

Given $H_0: \lambda = 1$ vs $H_1: \lambda \neq 1$, for $X_i \sim Poi(\lambda) i.i.d.$ construct the Score (Lagrange multiplier ) test. Note that the MLE is $\hat{\lambda} = \frac{1}{n} \sum_{i=1}^...
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19 views

Is the event independent?

I wanted to know if this event is independent or not? The question is as follows: In monsoon season, what is the probability that we will have a sunny day after a series of rainy days, or the ...
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1answer
29 views

What is the probability of an event happening after n series of times [closed]

I am learning to code in python and so for academic reasons i have decided to code this project. What tests should i do which will answer this question? In monsoon season, what is the probability that ...
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1answer
25 views

Likelihood Function and Maximum Likelihood Estimator for Random Variable dependent on another Geometrically Distributed Random Variable

Let $T_j$ be the year employee $j$ quits a job and it follows a geometric distribution with parameter $\theta$ such that $$ P(T_j=t)=\theta(1-\theta)^{t-1} $$ From a sample of $m$ employees, the ...
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1answer
51 views

Prove that $\frac{(n-2)s^2}{\sigma^2}\sim \chi^{2}_{n-2}$

Consider the following simple linear regression model involving the $\epsilon_i$ error term, $$y_i = \alpha + \beta x_i + \epsilon_i$$ such that, $$\epsilon_i \sim \mathcal N(0,\sigma^2)$$ we know ...
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Stuck on this probability question

Team A leads by 3 games vs. team B and 4 games vs. team C. Of the 9 remaining games, Teams A and C will play each other 6 more times and other teams 3 more times. Team B will play all 9 remaining ...
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How can I show that $X_n=e^n I_{\{Y>n\}} \to 0$ in probability?

Let $Y$ be a continuous random variable with density function $f_Y(y)=e^{-y}, y > 0$. Consider the sequence $\{X_n\}$, given by $X_n=e^nI_{\{Y>n\}}, n =1,2,\cdots$ How can I show that $\{X_n\} \...
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Evaluate an integral using importance sampling

Estimate $\int^{1}_{0}e^{x} dx$ using importance sampling. Should I use beta distribution as proposal distribution and uniform distribution as target ?
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27 views

Conditional expected value when conditioning on a set of values

Consider the random variables $Y,X,Z$ with supports $\mathcal{Y},\mathcal{X},\mathcal{Z}$, respectively. Suppose that $E(Y|X=x,Z=z)=0$ for each $(x,z)\in \mathcal{X}\times \mathcal{Z}$. Does this ...
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14 views

Does Standard Deviation Increase/Decrease when Average Increases?

I have a mean(88) and standard deviation(6) of sample data from period 1, I have the mean(91) for period 2 data; the mean increases from period 1 to period 2. Additionally, I have sample data from ...
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14 views

Addings noise to emperical distribution

Say I have the following samples from a joint distribution (note I only have access to the samples and not the functional form of the distribution it self) ...
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Different ways to analyze football (soccer) teams to predict the winner in the premier league

I am writing a 12-page high school paper on how to predict the winner of a football league(English Premier League). The statistics being used are goals scored home and away the previous season (I can ...
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Distinct membership [duplicate]

Let's say you have 10 groups of members. Each group can have an arbitrary number of members, but let's keep it clean (unless it matters?) and say each has 10,000 members. Between the combined ...
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How to optimize a number of games with a limit set of second chances at guessing other options given probabilities?

Let's say I have a game with $M$ outcomes $\{O_i\}_{i=1}^{M}$ and I have a predictor $P$ that gives me probabilities for the outcomes of a game. I now have $N$ independent games that I want to ...
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24 views

Infinite-sided dice riddle incorrect solution

Apologies for asking this as a new question instead of a comment on the other post: I don't have enough rep to comment... I was trying to solve this question: How to equalize the chance of throwing ...
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26 views

Adjust probability to preserve correlation

I'm doing my final project in psychology about correlation perception and want to implement this experiment: Let's say we have two variables - coffee consumption and alertness (with values: yes or no;...
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1answer
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What is the expected score of this game?

Construct a non-increasing sequence of integers between 0 and 9 by doing the following: 1) Draw a random integer between 1 and 9. Call this $X_1$. Multiply this number by $10^{-1}$. 2) To the ...
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How to equalize the chance of throwing the highest dice? (Riddle)

I just invented the following riddle, doing statistics work. (I actually need the answer!) Riddle: Imagine a dice game with the aim of throwing the highest dice. The dice are special and have ...
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2answers
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Consistency of estimators vs sample size

I understand that consistency of an estimator is large sample property, but does it make sense to talk about consistency in small samples as well? Can I say about the estimator that it is consistent ...
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Probability of winning a prize bond and the expected winning

In Pakistan, the government has a prize bond scheme that many citizens are investing in. I am a bit curious as to how can I calculate the odds of winning. I have some knowledge of statistics but my ...
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How to compute the probability of a sentence with neural network language models

Many introductions to language models in NLP start by saying that the language modeling task aims to find the probability of a sequence of inputs (typically word tokens or characters). For instance ...
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Calculating cdf given the probability density function

I'm trying to work out the cdf given this probability density function: $$f(x) = \cases{ 0.25 & for 0 < x < 2, \\0.5x+c & for 2 < x < 3,\\ 0 & \text{ elsewhere} }$$ The ...
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44 views

vector of three RVs that are pairwise Gaussian is Gaussian [duplicate]

If $(X,Y)$, $(X,Z)$, and $(Y,Z)$ are all Gaussian, does it follow that $(X,Y,Z)$ is also Gaussian? I'm having trouble coming up with a counterexample...
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38 views

Probability of event

You average 3,000 customers per day, of which, on average, 54 are high net worth (HNW). If you have 200 sales associates, who average 15 interactions per day, what’s the probability that a sales ...
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How does type of convergence affect results in practice? [closed]

I've been exposed to many different types of convergence of random variables, namely: convergence almost surely convergence in the $r^{th}$ moment convergence in probability convergence in ...
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33 views

Asymptotic behaviour of order statistic $x_{(n-k+1)}$ when k is $n^{\alpha}$

I am interested in the asymptotic behavior of the top k-th order statistic $x_{(n-k+1)}$ from n i.i.d. standard normal samples, when k is $n^\alpha$ where $\alpha\in (0,1)$. I just wonder if we can ...
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1answer
47 views

Why density plot tails are beyond maximum and minimum values?

I am trying to interpret the tails of a density curve, which go beyond xlims(0 in this case). I understand that area under the curve between any two points represents the probability of that event. ...
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What's the latent random variable that explains Polya's urn?

According to De-Fenitte, exhangeable r.v's can be thought of as iid given a latent random variable. My question is: for a Polya's urn model, what would this hidden random variable be?
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1answer
84 views

Statistical modeling multidimensional discrete system

I have a system which its state is described by a vector $v=(a, b, c)$, where $a$, $b$ and $c$ can take any value between $0$ and $100$ and where $a+b+c <= 100$. I have observations of the state ...
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Levy's Theorem (a form of CLT for random variables with infinite variance) [closed]

Possible extension to Levy's Theorem e.g., Theorem 3.8.2 of Durrett (p 186, version 5 of ``Probability: Theory and Examples"). Theorem 3.8.2 of Durrett (p 186) is, roughly speaking, a kind of form ...
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1answer
48 views

Union of intersection of B and A and intersection of B and A complementation

Due to my little knowledge in set theory, I simply don't know how the authors of Statistical Inference could make this highlighted statement Could someone please explain? What book should I read to ...
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1answer
47 views

Poisson-Gamma conjunction - calculating posterior [duplicate]

How to calculate posterior distribution step-by-step while given: some observed numbers of customers from the last days that number of clients is distributed by Poisson($\lambda$) ($\lambda$ is not ...
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3answers
818 views

How to calculate number of sets in Sigma Algebra

The example 1.2.2 of the book Statistical Inference by Casella and Berger states: if S has n elements, there are 2^n sets...(please see attached). Could you please explain how the authors derived ...