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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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Comparing observed vs expected occurrences of events when each occurrence has a different probability

I am trying to determine whether a set of advertised probabilities are accurate or if the advertiser is providing misleading data. Specifically, I have a list of events where the probability of ...
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Manipulating samples with Extreme Value Theory

First of all, i'm an electrical engineer and i know very little about statistics and probability. In one of my work, i'm trying to somehow combine(this word might be wrong..) Extreme Value Theory and ...
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Measure modality of a distribution

In statistics, different orders of moments are tools to characterize a distribution, for example mean, covariance, skewness etc., which also gives an intuitive way to visualize the distribution. But ...
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How to calculate number of solid color images on the screen of random pixel generator?

Is there a statistical formula/function to calculate number of basic geometric shapes on the displays of random pixel generator? Let's say we have a basic random pixel generator which has 10*10 ...
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Effect of a class to another class's passing rate

I have a case where there are 2 classes and I want to figure out how class 1 effects the passing rate of class 2. An example case: ...
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Bayesian hypothesis testing and NHST - Which is better

i've seen a lot of approach for hypothesis testing and i wonder if some test (mainly those coming from frequentist inference) like the null hypothesis testing are still relevants today. I've seen some ...
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31 views

Confusion with Computing Probabilities of a Normal Distribution without the Integral

How does this code is calculating the probability of Normal distribution without calculating the integral ...
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1answer
21 views

Definition of symmetric probability mass function [duplicate]

Consider the random variables $Y$ characterised by a probability mass function (PMF) as follows: $$ Y=\begin{cases} -2 & \text{ with probability $\frac{1}{2}$}\\ 2 & \text{ with probability $\...
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Prove limiting distribution goes to stationary distribution: $\lim_{t\to\infty} \pi_{j}(t) = \overline{\pi}_{j}$

This is a problem I'm struggling with on continuous-time Markov chains. Here, we are considering a continuous Markov process with phase space $\{1, 2\}$ (there are only two states). Moreover, $\...
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Show that the intersection of two sets involving symmetric PMF is empty

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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Manually zero bounding a metric before significance testing in an experiment

I’m trying to replicate the methodology described in this paper to improve the sensitivity of a significance test on a metric (number of purchases) in my experiment on an ecommerce website. In short, ...
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Best way to test the difference in means for Tweedie distributions?

What is the best way to test the difference in means for two samples of data that follow a Tweedie distribution? (zero inflated, tweedie power parameter ~1.7) The data represents the number of ...
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Characterise the set of symmetric probability mass functions

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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Likelihood deviation on diseased test

I feel confused with the following problem: the population $n=237$ diseased people are required to perform a 6-successive-day diagnostic test on cancer. And the random variables are given as follows: ...
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1answer
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How is Student's t-distribution related with this similarity/probability equation between data points?

In the t-SNE paper "Visualizing Data using t-SNE" and a Deep Embedded Clustering (DEC) approach "Unsupervised Deep Embedding for Clustering Analysis", they both use the Student t-distribution to ...
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Why is this a valid step (expectation w.r.t posterior)?

Reading through this paper and on page 10 they use the step: $$\int q(\theta|D,\phi) \log p(Y|X,\theta) d\theta = E_q \log p(Y|X,\theta)$$ Now obviously I understand why they have written this as an ...
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Probability for effect of intervention [closed]

Say that 40 % of patients with disease X recover without treatment. With a specific treatment 80% of the patients recover. You are going to give the treatment to a randomly chosen patient with this ...
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How many neurons are actually dropped when using dropout?

I understand that when using dropout, a single neuron can be described using Bernoulli random variable and for a set of neurons it can be described as Binomial random variable When using Dropout, we ...
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Sum of N random variables from the same distributions [duplicate]

Given $n$ independent random variables from the same distribution, how to obtain the distribution of their sum? For example, the distribution of $n$ normal distribution is $N(n\mu, n\delta^2)$. ...
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1answer
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Summing Posterior Probability of Naive Bayes

I'm trying to work through a toy example of Naive Bayes with text classification (spam/ham) to make sure I understand the intuition, but not understanding why my posterior probabilities are not ...
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Can we apply Gaussian mixture model to all kinds of scensrios where some variables are unobserved?

I learned from this answer that: A mixture distribution combines different component distributions with weights that typically sum to one (or can be renormalized). A gaussian-mixture is the ...
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Birthday problem extension to unequal probabilities and multiple collisions

Let $p_1, ... ,p_k$ denote the probabilities of drawing bin $1, .. ,k$, where $\sum_{i = 1}^{k} p_i= 1$. My question is if we draw $n$ times, how can I show that the probability of no bin being drawn $...
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Bounds on Expectation $E[A(B-C)^2]$

[This question has been edited for more given conditions]. Given possibly correlated random variables $A,B,C$, I want to find the best upper bound for $E[A(B-C)^2]$ given the following: $E[A(B-C)]$ $...
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1answer
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Recursion in Probability questions

I don't quit understand the explanation for recursion here. Why is there firstly the need to have i on the numerator for the first underlined bit? Would it be because of combinatorics? So the second ...
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Calculate E[X] from incomplete data?

The exercise I'm doing describes the random variable $X$ as the following ...
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1answer
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Practical but pretty basic “real world” statistics problem

It was suggested that this forum may be quite helpful. My most humble apologies if I am somehow not posting in the right place! My bosses asked me (I am not a statistician) how many letters have to ...
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Setting an A/B test for electronic devices after an update

I want to set up an experimental testing or A/B test. We have electronic devices on the field. Recently we had a problem with the keydoor. It started falling(it wont open(Fail) after we try to open ...
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1answer
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How to decide which Probability distribution to use on a specific problem?

I need some guidance. Which probability distribution I can use to understand the number of orders one person can handle in a company? It's like each person has orders which takes minimum of 50 days to ...
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Variation on the Birthday Problem

You ask an audience one by one for their birthdays. How many people do you need to ask on average until you get your first overlap? This sounded to me quite similar to a geometric distribution in ...
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Numerical examples proving and disproving the optimal scaling heuristic by Roberts et al

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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Transition probability in Markov chain using a decision tree model

I wish to find a way to calculate the transition probabilities in my Markov chain model. Let's say a customer has three products [A B C] and in this example I wish to know the transition probability ...
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Are data considered to be “events” or “random variables” in machine learning?

I was sitting at a lecture on Naive Bayes, and the speaker, on a slide, said: Given a feature $x = \begin{bmatrix} x_1, \ldots, x_n \end{bmatrix}^T$, the probability of the feature belong to class ...
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1answer
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Making predictions in Logistic regression in R

Can anyone please let me know, what does a logistic regression model predict? For example, there are 2 variables. This may not be very appropriate example but it will clarify my question as follows: <...
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1answer
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What happens if my Anderson Darling score is more than 10?

I am using crystal ball to fit a distribution curves. Based on the fitting, the AD score is 20, p value is 0. What does it mean? Does it mean it is suitable? Appreciate any help! Thanks!
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About basic terminology: experiment, outcome, sample space, is my understanding correct?

Part 1 We could define "tossing $1$ coin $1$ time" as an experiment $E_1$, with possible outcomes, aka, the sample space of which is $\{H,T\}$, this set has 2 elements; "tossing $1$ coins $2$ ...
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LDA visualization:

I am trying to find ways to visualize topics(LDA) to test something I am working on. I used to use pyLDAvis in python but currently what I am working on outputs only two items: 1) P(Topic | word), ...
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1answer
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Why $P(BB|BG \cup BB) = \frac{P(BB)}{P(BG \cup BB)}$?

I have the following question: Determine the probability both children are boys if I have 2 children, the elder is a boy? Then the textbook says: Take BB, BG, GB, GG as the possible outcomes (first ...
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Expected number of visits to the state in a Markov chain

We know that in a Markov chain the taboo probability, $P^n_{i,i+1}(H)=Pr\{ X_n=i+1,X_k$ doesn't belong to H for $0 < k < n| X_0=i \}$, where H is an arbitrary set of states in a Markov chain $...
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1answer
29 views

Let A and B be independent. Show that $A^c$ and $B^c$ c are independent - better solution?

My solution seems too long winded. This should take less writing, no? Here it is. $B^c$ means complement of $B$ (i.e. $P(B) = 1 - P(B^c)$ If $A$ and $B$ are independent events: $$ P(A \cap B) = P(A)...
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2answers
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Distinction between probability and probability mass function (PMF)?

Assume $X$ is a discrete-valued random variable. Often in literature the same notation is used for the probability $Pr(X = x)$ as for the probability mass function $P(X)$. Clearly, this is because ...
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How to make machine learning model that reports ambiguity of the input?

Suppose I want to build a neural network regression model that takes one input and return one output. Here's the training data: ...
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The expected occurences of successive draws

We throw a coin 1 000 000 times. How many times on average will make 13 successful heads? Now the problem with the naive: 1 000 000/(2^13) is that once it made 13 heads the 14 head will happen with 1/...
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How to create a scale combining normalized percentile scores to get an overall result

I am trying to work out a scale for assessing a person’s risk for a particular behavior (like rob a bank). The likelihood of this behavior occurring or not occurring will be based on scores for a ...
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Capital distribution between several bets

I'm trying to figure out the algorithm of capital distribution between several bets. For example, let's say I have two bets with the same 50%/50% probability of winning and losing, but one bet has a ...
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2answers
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How is $P(D;\theta) = P(D|\theta)$?

I started reading about maximum likelihood estimator and Bayesian statistics recently. I understand that given a statistical model $(X, (P_\theta))$ where, $\theta$ belongs to a large parameter space $...
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Terminology Confusion: Probability, Likelihood, AND?

currently I have a terminology issue to accurately write a text. I read up on the definitions on probability and likelihood. Given some continuous random variable(s) as far as I can infer, probability ...
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Find the same components between two distributions

GMM assumes that a distribution consists of multiple Gaussian Distributions. Is it possible to find the same components between the two distribution? For example, there are two distribution D1 and D2. ...
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How can I have probability decrease linearly with more dice?

I'm working on a simplified rpg-system that only use D6 and I want a mechanic for fumble/critical fail. Depending on how good the player is they have 1-5 dice to roll and they have to beat a ...
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How do I calculate a Bayesian Posterior Distribution from an Exponential Prior and Sample Data

I have a dataset where each observation is a length of time (e.g. 50 days, 70 days, 105 days) and I am trying to utilize Bayesian statistics to calculate a posterior distribution in light of new data. ...
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2answers
72 views

Probability Theory and distribution [closed]

I am studying probability theory and one of the questions that I have faced is this. The problem is that I either don't know where to go about with this question or even if I do do something about it, ...