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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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Interpretation of the rate parameter of a Gamma distribution

I am toying with mixture models, especially in a bayesian context and the Gamma (or the inverse Gamma) distribution appears quite often. For example, inverse Gamma is used as a conjugate prior for the ...
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1answer
22 views

Why do we care about the joint distribution of the endogenous variables of a causal model?

In general, we can calculate the joint distribution of the endogenous variables of a structural casual model (SCM) as follows $$ P(X, X_1, \dots, X_n) = \prod_i = P(X_i \mid \text{parent}(X_i)) $$ ...
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1answer
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Estimating Differential entropy from unbiased samples of a probability distribution

If I can get N unbiased samples $x$ from $p(x)$ how can I approximate the Differential entropy: $$H(X) = -\displaystyle\int_{x} p(x)\log p(x) dx$$ I'm not very knowledgeable in statistics so I'm not ...
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How can I model flips until N successes?

You and I decide to play a game where we take turns flipping a coin. The first player to flip 10 heads in total wins the game. Naturally, there is an argument about who should go first. Simulations ...
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PDF of $\frac{(X_1-X_2)^{2}}{2}$ where X1 and X2 are independent standard normal [on hold]

I'm trying to find the PDF of $$\frac{(X_1-X_2)^{2}}{2}$$ where X1 and X2 are independent standard normal. Any hints on how to proceed with this?
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Probabilistic user behavior markov models on web

I am considering the following probabilistic Markov model of actions of a user on the results page of a search engine. The user examines the first result, with a probability $A$ he is satisfied with ...
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4answers
49 views

How to build a Bayesian Model to estimate the probability distribution of the parameters given the output?

I'm currently facing a new type of problem, and i have no idea how to solve it, so any suggestion will be really appreciated ! The problem is the following: I have a matrix of temperatures, depending ...
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11 views

Goodness-of-fit test when the sample space is monotonically increasing

I came across this paper which develops a GoF test on data drawn from a circular sample space (i.e. it has cyclical support). I am now wondering if there is a parallel to monotonic supports (or if ...
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1answer
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doubtful regarding my solution to bonferroni's principle exercise

Self learning and not quite good at probablility and statistics, my question is regarding solution to exercise 1.2.1 in chapter 1 of Mining of Massive Datasets book. The text of the exercise reads: ...
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Questions on Notation for PMF and Expectation

Before diving into the Stanford CS229 Machine Learning notes online, I decided to go through the course's notes on probability review and had a few questions. In section 2.2, it states A ...
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Count probability of average and variance of random variable with normal distribution [on hold]

Please verify my calculation for four exercises placed below. Example 1. Random variable $X$ has distribution $N(100; 10)$. Count $P(90 < \overline{X}_9 < 95)$. Example 2. Random variable $X$ ...
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Is f(x) = 1/n for x = 0,1,2,…,n a valid PMF? (n>0 an integer) [on hold]

Is f(x) = 1/n for x = 0,1,2,...,n a valid PMF? (n>0 an integer) Intuitively thinking, the summation of the function over the range equals to (n+1)/n which is greater than 1 and so the function isn't ...
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1answer
83 views

Proving that given Markov chain is homogeneous. Find state space and transition matrix

Let $X_i$ be the results of a consecutive throws of a die. Let $Z_n=3(X_1^2+\cdots+X_n^2) \bmod 5$. Show that the sequence ${\{Z_n \mid n\geq1\}}$ is a homogeneous Markov Chain. Find a state space and ...
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2answers
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Deriving a distribution whose pdf has the shape of a square + a triangle (a right trapezoid)

I want to the derive the PDF which looks like the sum of a triangular and uniform distribution which looks like this: To do this I have simply added the PDFs for the rectangular and triangular parts, ...
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Probability of userbase recieving special page

Say I have an app. When the app is launched there is a process behind the scenes the determines which of the two outputs the user receives, default or special. The default output displays the app ...
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1answer
44 views

Proof of probabilities that may not be independent

I am given the problem: Given $P(A) = \frac{3}{4} $, $P(B) = \frac{3}{8} $, show that: a) $P(A or B) > \frac{3}{4} $. b) $\frac{1}{8} < P(A and B) < \frac{3}{8} $. The problem does not ...
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1answer
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On the predictive distribution regarding Bayesian methods for pattern recognition

I am following Pattern recognition and machine learning by Byshop and I was trying to derive myself the predictive distribution resulting from a new data point. For the sake of clarity I post the ...
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1answer
26 views

differences between conditional probability and dependency

Sometimes, I read articles about conditional probabilities and other articles about conditional dependency. My question what is the main differences between them? For example, "https://en.wikipedia....
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25 views

Independent Study Statistics/Probability Grad Level [duplicate]

I am trying to decide on topics for my independent study this semester. I am a Pre-Doctoral Mathematics student, so looking for a more math based text rather than engineering based (which I have found ...
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1answer
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Optimal scaling of the Random Walk Metroplis-Hastings algorithm and the speed measure of the limiting diffusion

Let $d\in\mathbb N$ with $d>1$ $\ell>0$ $\sigma_d^2:=\frac{\ell^2}{d-1}$ $f\in C^2(\mathbb R)$ be positive with $$\int f(x)\:{\rm d}x=1$$ and $g:=\ln f$ $Q_d$ be a Markov kernel on $(\mathbb R^...
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1answer
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What is probabilistic inference?

I am reading Chris Bishop's Pattern Recognition and Machine Learning textbook. I came across the term probabilistic inference several times. I have a couple of questions. Is probabilistic inference ...
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Co-occurence in a population

Let's say I have a population of size N that has traits a-z. Some people/samples might have just a,n,x, etc... P(a), ..., P(z) is known. # of people with i & j in the population is known for all ...
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States of Markov chain and stationary distribution

Let $X$ be a Markov chain with a state space $S={\{0,1,2,... \}}$ and a transition matrix $P$ with given $p_{i,0}=\frac{i}{i+1}$ and $p_{i,i+1}=\frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states ...
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Probability that exactly 12 buses will arrive within 3 hours

Let's suppose there are two buses $A$ and $B$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the ...
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8 views

Series Winner Probability

Suppose we have Series 1 and Series 2, and four teams $A,B,C,D$. $A$ and $B$ play in Series 1 and $C$ and $D$ play in Series 2. In each series, the first team to win 3 games wins the series. The home ...
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0answers
14 views

Probability Winning Game [duplicate]

Suppose you are tied $1-1$ with Bob. To win the game, you have win greater than or equal to $5$ rounds and win at least $2$ more rounds than your opponent. The probability of winning a round is $40\%$ ...
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1answer
81 views

Would you flag this data as fraudulent?

Let's suppose you have been given some data from a randomized block design with 4 repetitions and 23 treatments. After an initial inspection of the data, you notice that for 8 treatments all ...
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2answers
62 views

How can we best explain causality for the uninitiated?

How can we best explain causality in layman's terms? There seem to be two main types of causality. One is probabilistic causation, the other is called determinism in philosophic circles or just ...
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3answers
857 views

Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...
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0answers
38 views

Find the distribution function $F$ for $min_{1 \le i \le n}{X_i}$ [duplicate]

Given a random sample $X_1, X_2, ..., X_n$ where each $X_i$ has pdf: $$ f(x; \theta) = 3 \theta^3 x^{-4} $$ and $0 \lt \theta \le x \le \infty$. Show that the distribution function $F$ for $min_{1 \...
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Calculating the probability of winning a game of Rainbow Six Siege

I'm currently developing an application in Python that calculates the probability of winning a game of Rainbow Six Siege based on the skill level of the players in the game. Siege is a 5 vs. 5 ...
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Sufficient conditions for multivariate MGF to be finite in neighborhood of zero

In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof) Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
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0answers
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NFL Playoff Game Probabilities

Context: My friends and I have this homebrew NFL fantasy football game we play for the playoffs once the NFL regular season is over. The contest is not head-to-head with other players like standard ...
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2answers
81 views

Analysis concepts relevant to probability theory

I am taking a course based on Durett. I was wondering if anyone could point me to the analysis skills or topics that are most relevant to Probability Theory. It has been some time since I've taken ...
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2answers
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What does $\sum_{\{\textbf{x}:T(\textbf{x})=t\}} f(\textbf{x}; \theta)$ mean?

This is in the following context: $$ q(t;\theta) = P(T=t;\theta) = \sum_{\{\textbf{x}:T(\textbf{x})=t\}} f(\textbf{x}; \theta) $$ Where $T=T(\textbf{X})$ is a statistic, $q(t; \theta)$ is the pmf of ...
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0answers
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Is the sum of one slowly varying and one regularly varying random variable regularly varying?

Imagine I have given a random variable $X$ with support $\geq 0$ and $\mathbb P(X=0)>0$. Also, I have another independent random variable $Y$ with positive support being regularly varying at zero ...
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Is the sum of two dependent sub-gaussian variables X and Y still follows sub-gaussian distribution?

I am trying to prove the random vector with dependent sub-gaussian coordinates is also a sub-gaussian random vector. The related question has been asked in https://math.stackexchange.com/q/3072363/...
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1answer
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How to find the value for average rate

I'm doing some textbook problems on my own and there are no steps given to the solutions to some of the problems. If anyone could help me solve the following problem, much would be appreciated. "A ...
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3answers
9k views

Are Event and Outcome synonymous?

Outcome : An outcome is a result of a random experiment. Event : A single result of an experiment. Are Event and Outcome synonymous ?
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1answer
23 views

Recursive Bayes Learning

I'm trying to work through an example from Richard Dudas Pattern Classification on Recursive Bayes Learning. My main question is why do we choose the $max[D^n] $ in: $$max[D^n] \le \theta \le 10 $$ ...
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1answer
28 views

Compute $P(2\leq x\leq 8)$ with Poisson distribution and $\lambda=7.2$

Compute $P(2\leq x\leq 8)$ with Poisson distribution and $\lambda=7.2$ My attempt: I need calculate this using $R$. then I use this: ...
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0answers
13 views

Writing down a conditional probability / finding the underlying probability space

This is a notational issue: Let's say we have a partition of $\mathcal X$ given by $\{\Omega,\Sigma\}$ and we define $P(\omega\in\Omega) = p$, $P(X(\omega)=x|\omega\in\Omega) = \pi$ and $P(X(\...
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0answers
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Probability with or and under in the questions [closed]

Can anyone please help me to solve these problem with explanation as possible.
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1answer
141 views

Large deviations proof question

Below is part of the proof of large deviations result. K is cumulant generating function. Can anyone explain how the last step follows? This is page 157 of McCullagh's "Tensor Methods in Statistics"
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1answer
27 views

Probability of finding a lost item

I am trying to solve the following problem and was wondering if someone can verify my answers. Big Joe has lost an important document. There is a 70% probability it is at home, and a 30% chance it is ...
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1answer
98 views
+50

Probabilistic Interpretation of Radial Basis Function

I was wondering if someone could flesh out the probabilistic interpretation of using the Radial Basis Function to compute the probability between an observation and some reference value. My question ...
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2answers
36 views

Permutation or Combination?

I am a bit confused, if I have 2 football teams, and the results are three (1: home win, 2: away win and X: draw), I can see that the possibilities for these teams results are 9, which formula (...
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0answers
91 views

Probability Mass Functions [closed]

In a bowling "frame", between 0 and 10 pins might be knocked over by a bowling ball. Suppose that a particular bowling alley attracts a mixture of beginners and experts, such that we can assume that ...
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0answers
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probability sampling for retrospective chart review RCR/Medical Record Review

how best can you achieve probability sampling in a medical record review. In answering question, please use a real example. Say you have 4000 pregnancies to review, how many of these do you need to ...
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Classic probability for contradicting information

We are currently trying out a statistical approach for user attribute confidence on user data, based on different sources providing this data. As an example, let's say we build this initial ...