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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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confidence interval of function of parameter

Self Study Question: We already know that the confidence interval of a parameter M is (-1,2). We are supposed to find the confidence interval of 1/M. The function is 1/X, which is One to One, so ...
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The expected outcome of a random distribution of marbles among several people

The problem is: On each round of a game, 20 marbles are distributed at random among five children: Alan, Ben, Carl, Dan, and Ed. Consider the following distributions: I) Alan: 4, Ben: 4, ...
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Probability of matches being random

We have a set of 576 samples that have been assayed to obtain some omics information (genomic/epigenomic). The 576 samples are mostly from unique individuals, but we have some samples for which the ...
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Definition of CDF of discrete RV

In many different (serious and good) statistics books I find different definitions of CDF of a discrete RV. The difference is the equal sign at the index of the summation sign. The first is: $$F(x) = ...
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An urn contains n unique marbles. I draw k of them with replacement, and get r duplicates (r>0). What is the MLE for n?

What is the MLE for n which is the total number of marbles in the urn? What is the expected value of n? And what if r = 0 (no duplicates were drawn)? Can anything be said about n? A duplicate ...
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Finding $P(X>d)$ when $X\mid \Lambda$ is Pareto and $\Lambda$ is Gamma distributed

Supposedly $X$ have a Pareto distribution with parameters $\Lambda$ and $\theta$. Let $\Lambda$ have a Gamma distribution with parameters $\alpha$ and $1$ (i.e., scale parameter $= 1$). Calculate (...
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Derive multivariate from bivariate normal distribution

Could anyone help me on the following. Let $K$ and $M$ be integers so that $K\geq3$ and $2\leq M < K$. Let $\boldsymbol{X}=(X_1, ..., X_K)^T$ be a random vector, $\boldsymbol{\mu}$ be a $K\times 1$...
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Derive the CDF of the sum of two independent random variables [duplicate]

Some notational remarks before presenting the question: $k<\infty$, with $k\in \mathbb{N}$. $\lambda\equiv (\lambda_1,...,\lambda_k)$, $\lambda_j\in [0,1]^k$ $\forall j$, and $\sum_{j=1}^k\...
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Do you need to adjust the probability if you use the 'class_weight' parameter in LogisticRegression-sklearn?

I have a imbalanced dataset and I want the the output as probabilities and not labels. Hence using Logistic Regression seemed to be the obvious choice. However the classsifer started predicting all ...
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selecting a number according to probability

Could anyone just tell me what does he, mean by at the point $2$, selecting a number according to the probabilities, $p(x_0)$? does he meant to chhose the largest one out of $p_i(x_0), i=1,2,\dots,N$?,...
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Discussion on a probability question from Puzzling.SE

BACKSTORY AND RELEVANT LINKS So over on Puzzling.SE, a user asked this question. DISCLAIMER: I'm not asking you guys for the answer to the question, but rather an answer to the discussion that ensued ...
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Showing that if the PMF of $W$ is symmetric around zero then some parameters entering it are equivalent

Summary: In what follows, I specify the probability mass function (PMF) of a random variable $W$, depending on some parameters $(\lambda,\mu,\lambda',\mu')$. I would like your help to show that $$ \...
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1answer
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Sample space for 4 consecutive coin flips vs 4 coins flipped at once

I am trying to firm up my understanding of sample space. I am playing with the idea of two random processes, where one realization of each process is (A): flip a coin 4 times, record the results (B):...
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Discriminant functions vs. non-Discriminant [closed]

How can I prove which ones from the following functions are non-Discriminant and which are? is it all Bayesian rule? can I use the joint rule in proving? x is the data w1 is the class/label P(...
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If a and b are dependent variables, what's the probability of a,b both happen?

Bayes' theorem tells me $$P(a \cap b)= P(a)P(b|a) \tag{1}$$ which I interpret as a happens, and b happens on condition of a. Then why don't we need to consider b happens first, ie. $$P(a \cap b)= ...
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1answer
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Does the peak of a Normal Distribution mean anything? [closed]

What does the peak of a Normal distribution show? Let's say if I have a flat peak, does this mean I have a larger variance? What if I have a sharp peak? For example, Does the "blue distribution" ...
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Expected value of a variable given a probability distribution

I have fitted some data with a normal and gamma distribution. But I need to find the expected value of my variable. For example is x is my data (mine is not normal) and fit them into a normal and ...
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Bayesian (In)Decision

Let $A_j$ be the action of person $j$, $A_k$ be the action of person $k$, and $p(A)$ be the probability of an action. Using Bayes Rule, $$p(A_j=x|A_k=y)=\frac{p(A_k=y|A_j=x)p(A_j=x)}{p(A_k=y)}$$ If $...
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What is the $p$ in Bernoulli distribution?

In the Bayesian theory of probability, probability is our expression of knowledge about a certain thing, not a property of that thing. However, I always see people treat $p$ as a parameter that needs ...
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Calculating Conditional Probability in a small cipher problem

I have a small part of data that is n bit long (0 or 1). Probability that a bit is 0 equals p. I also have a key that is used to cipher this data that is 1 bit long. The key will be 0 or 1 for the ...
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Finding unconditional pdf of a Pareto-Gamma mixed distribution

Let X have a Pareto distribution with parameters Λ and θ. Let Λ have a gamma distribution with parameters α and 1 (i.e., scale parameter = 1). Find the unconditional pdf of X. I tried finding the ...
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Robust Trimmed Mean Proof

This question arises from the first chapter of "ROBUST LEARNING: INFORMATION THEORY AND ALGORITHMS" by Jacob Steinhardt. Given is the following: there are $\epsilon n$ outliers and $(1-\epsilon)n$ ...
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Probability of detecting small bias in a die in the low confidence regime

We are given a biased $m$-sided die: one of the sides has probability $\frac{1}{m} + \gamma$ and all the rest have probability $\frac{1}{m} - \frac{\gamma}{m-1}$ each. The goal is to figure out which ...
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Binomial distribution intituition for N

I am unable to convince myself intuitively as to why the variance of a binomial distribution increases with increase in n (number of trials). In general, I expect that as n increases, the distribution ...
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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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1answer
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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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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
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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 ...