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Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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7 views

Calculation of the n-th central moment of the F distribution $F(m,m)$ [on hold]

Calculation of the n-th central moment of the $F$ distribution $F(m,m)$.
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0answers
13 views

What is the distribution of the peak time of the first hitting time process

I need to find the distribution of the random variable $T_{peak}$ where $T_{peak}$ represents the peak time of the first hitting time process. Detailed Explanation of the System: There are emitted ...
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0answers
21 views

Joint distribution function

Suppose that we have two indpendent random variables $X, Y$ with a joint probability density function $f(x,y)=1$, $-y<x<y$, $0<y<1$ How can I calculate the cumulative joint probability ...
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1answer
36 views

If all trimmed means are equal does this imply equal distributions?

I am trying to prove the following: Given that $\forall \alpha\in [0,1]$: $$\int_{F_S^{-1}(\alpha)}^{\infty}xf_S(x)\,dx = \int_{F_0^{-1}(\alpha)}^{\infty}yf_0(y)\,dy$$ where $F_S^{-1}(\alpha)$ and $...
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2answers
28 views

How can I calculate the probability of a increasingly likely positive outcome?

I'm not even sure I'm phrasing the question properly, please let me know if there is any standard terms around this type of problem. I'm trying to find an average number of attempts it would take to ...
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0answers
18 views

Test to determine whether the empirical distribution for a given day is an outlier compared with other days

Say you have multiple data samples from different days (or some other unit of time) and you want to answer the question: is the distribution for a given day an outlier (compared to other days)? Is ...
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2answers
50 views

Why use histogram to illustrated probability distribution

Forgive me I am a newbie of random variables. I saw a lot of course which introduce the Discrete Random Variables which always be illustrated with a histogram or ...
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0answers
13 views

Mutual info between continuous and discrete variables from numerical data

I am looking for references/measures to estimate the mutual information between a continuous (C) and discrete (D) variable, given a real-world (i.e. finite sample) data set. C is uniformly distributed ...
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1answer
287 views

Difference between function and distribution?

I know this is dumb question, but i am confused to understand it. I have constant function as $$y = exp(-x^2)$$ This also represent gaussian distribution with mean zero and variance of 0.5 Now if ...
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1answer
38 views

Random Variables and Probability

So I encountered this problem while I was studying for exam. However, I cannot wrap my head around the solution that the answer key provided. The problem goes like this: Bob watches cars that pass ...
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1answer
14 views

Probability that one random variable using the Beta Distribution being greater than another, bounded intervals

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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1answer
19 views

How to fit a superimposed distribution (\eg a Gaussian distribution + a Uniform distribution)

Suppose we have a set of independent observations of a random variable X, which is a Superimposition of two mutual independent random variables (i.e. X = Y + Z), Y follows a uniform distribution, ...
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1answer
27 views

What does it mean to “interpret the sigmoid $\sigma(\theta^Tx)$ as a probability”? [duplicate]

In Goodfellow's Deep learning text, it is written Is this way of defining a probability $p(y=1| x;\theta)$ even legal? Recall the definition of a probability given a random variable where $p_X$ is ...
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0answers
15 views

Distribution of the average of multivariate normals?

I have seen that the sum of $n$ iid multivariate normal vectors (mean $\mu$ and variance $\Sigma$), $X_1+\dots+X_n$, is distributed as a normal with mean $n\mu$ and variance $n\Sigma$. Is the ...
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0answers
32 views

What is the expectation of Poisson arrival times? [on hold]

I am interested in the expected wait time (till time $t$) of Poisson arrivals, $$\frac{\sum^{N(t)}_{i=1}{(t-t_i)}}{N(t)}$$ where $t_i$ is the arrival time of customer $i$, $i\in[1,N(t)]$. It is ...
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0answers
35 views

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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1answer
23 views

Help creating a formula for a probability problem

I was thinking of a probability problem, but I'm having trouble thinking about what the formula would be to solve it. Say you need 10 events to win a game. 10% of the time you press a button, it ...
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0answers
20 views

How to compare two sets of unequal data

I am doing a users perception research that compares experts opinions and novice users opinion for an application. The data collected was qualitative data from focus group. After analysing the ...
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0answers
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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0answers
15 views

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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0answers
26 views

Is f(x) = 1/n for x = 0,1,2,…,n a valid PMF? (n>0 an integer) [closed]

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
20 views

Symmetric probability density function proof [duplicate]

The problem is stated as: Let $f$ denote the density function of the random variable $X$. $X$ has a symmetric distribution around $a$, in other words, $f(a+h) = f(a-h)$. Prove that $E(X) = a$, ...
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1answer
28 views

Continous random variable short proof

I am given the problem: If X is a continuous random variable with cumulative distribution function F and density function f, show that the random variable Y = X^2 is also continuous and express its ...
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4answers
60 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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1answer
33 views

Bayesian posterior pmf for weighted dice with uniform prior

We want to find posterior probability mass function for dice tossing with uniform prior. We are interested in rolling of weighted dice. The outcome is 1,2,...,6. We assume that prior probability ...
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2answers
37 views

Does an inequality hold as an expectation over a probability distribution?

Suppose I have to functions $f(x)$ and $g(x)$ such that $$ f(x) \leq g(x) \quad \forall x. $$ For a distribution $\pi(x)$ on $x$, is it necessarily true that $$ E_\pi[f(x)] \leq E_\pi[g(x)]? $$ My ...
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0answers
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R: Why does my Expectation-Maximization estimation for bimodial distribution give the wrong cutoff value? [closed]

I am putting together a regression model with data of carseat sales from the ISLR dataset. It is sales as a function of the independent variables. One of the variables has a bimodal distribution ...
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0answers
21 views

How do I Identify a cutoff value from bimodal data?

I am putting together a regression model with data of carseat sales from the ISLR dataset. It is sales as a function of the independent variables. One of the variables has a bimodal distribution I ...
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0answers
28 views

What is the Population? [closed]

So, in AP Statistics, we had the below question (we are in the sampling distributions chapter), which he decided to spring on us in a graded quiz with no previous examples similar to the below problem:...
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0answers
17 views

Validate Binomial distribution exercise with pullets and cockerels

The exercise is as follows: The owner of chicken farm knows, that the event of hatching out a cockerel is three time more frequent than a pullet. One should count the probability, that from five ...
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1answer
76 views

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
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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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0answers
19 views

Prior predictive distribution usage

I understand the mechanics and math behind prior predictive distributions, but I don't understand its practical uses. Theoretically and application wise, what is its purpose?
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0answers
15 views

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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0answers
10 views

Mutual info between discrete and continuous RV with history dependence

I am looking for literature references/measures to compute the mutual information between a continuous variable (time series) and a binary variable (a temporal sequence of 0's/1's). Briefly, a time ...
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1answer
18 views

Mixed distribution

Bob is a carnival operator of a game in which a player receives a prize worth $W=2^N$ if the player has $N$ successes, $N=0,1,2,3,...$ Bob models the probability of success for a player as follows: i)$...
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0answers
16 views

How to determine if a dichotomous variable is randomly distributed or is predicted by other instances

I have a product with a measurement of X features. Each feature can either be a PASS or a FAIL. Which statistical test can I use to tell if the failures are randomly distributed across all features? ...
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2answers
45 views

Inference based on a single observation

Imagine we use machine A to perform a task, we repeat it 1000 times and it always takes more then 30 min to finish the task. We buy a new machine (machine B) and in the first run it takes 29 min to ...
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0answers
25 views

Determining differences between curves with the K-S test

I'm running an experiment where my treatments have a significant and highly sensitive effect of the distribution of a result. For example, if i alter a certain nutrient in a vegetable, the ...
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1answer
12 views

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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1answer
109 views
+50

Bayesian inference for non-Gaussian errors

Following from a previously unanswered question, regression tasks involving measurements with normally distributed noise apply Gaussian processes. But are there any recommended approaches for ...
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2answers
77 views

Independence of Beta ratios of Gamma variates

If $X= x_1/(x_1+x_2+x_3)$ and $Y= x_2/(x_1+x_2+x_3)$ where $x_1, x_2, x_3$ are independent $\chi^2$-distributed random variables with d.f. $-n_1,n_2, n_3$ respectively. Are $X$ and $Y$ independent? I ...
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0answers
19 views

Likelihood function vs probability distribution function [duplicate]

I've been reading about Bayesian statistics and data analysis, and constantly see that $\text{posterior} \propto \text{prior} \ \times \text{likelihood}$. I'm familiar with fundamental probability and ...
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1answer
16 views

Combining forecasts or distributions to form a more accurate one

Suppose you are interested in getting as good an estimate as possible for a random variable $X$, this could be for example a stock price in the future. You go to see $N$ experts, each gives you a ...
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0answers
12 views

Histogram of Subtraction of Underlying Values

I have histograms produced from two sets of data recording. One of background noise of values $X$ and another $Z$ with a signal of values $Y$ present such that $Z = X + Y$. How can I estimate a ...
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3answers
37 views

Two Distributions, One a Sum: Discerning likelihood given error

Given $X, Y$ independent and non-normal, I'm recording histograms of $X$ and of $Z = X + Y$, sampled when $Y$ is not present and when it is, respectfully. I'm trying to figure out $Var(Y)$ and its ...
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2answers
106 views

Independence of ratios of independent variates

If $X= x_1/(x_1+x_2)$ and $Y= (x_1+x_2)/(x_1+x_2+x_3)$ where $x_1,x_2,x_3$ independent chi-square variates with d.f $n_1,n_2,n_3$ respectively, are $X$ & $Y$ independent? I know the condition ...
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1answer
14 views

Proving completeness of highest-order statistic using Leibnitz' Rule

Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$ I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
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1answer
22 views

RCT analysis using ANCOVA for rates

I have a question based on the following approach for the analysis of RCT's. The following works well for the outcome (and baseline) being continuous with normal errors. Expanding upon this, I was ...
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0answers
26 views

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 ...