Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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2answers
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Online density estimation and learning

Suppose that I have a system that at each time $t_i$ produces $N$ i.i.d samples of an unknown distribution $f(x;t)$. I want to estimate the distribution in an online manner. If I had only the ...
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1answer
32 views

What is the most fitting distribution for “how much time each day” kind of variable?

I frequently encounter variables with values restricted in a known interval but otherwise looking like being normally distributed. A typical one is, say, time spent browsing internet each day. I am ...
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Good way to quantify difference/distance between (discrete) distributions?

I have a model that outputs sequences based on a multinomial distribution. A sequence is an ordered list of one-hot samples. Each sample has its own discrete distribution. So for instance, a sampling ...
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1answer
26 views

How to find the best fit margins distribution to my data

I have data as follow: ...
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1answer
32 views

Exponents in Binomial Distribution

Why do we have exponentials in equation of binomial distribution? What's the intuition of it?
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11 views

Question about Functions of Several Random Variables

In the Mathematical Statistics and Data Analysis by John Rice, it states that for random variables $U,V$ which are functions of random variables $X,Y$, we have: We know that $$f_{UV}(u,v) = f_{XY}(h_1(...
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Published source for D-dimensional behaviour of Dot-Product

I am currently studying the behaviour of the dot product between two random vectors in $R^d$. Specifically I wanted to start with the case of uniform random vectors on $\mathcal{S}^{d-1}$. I found ...
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2answers
63 views

Convergence in distribution of sum of random variables

Let $\{x_{1,n}\}_{n\in\mathbb{N}},...,\{x_{k,n}\}_{n\in\mathbb{N}}$ be random sequences of zero mean random variables satisfying $$x_{1,n}\overset{d}{\to} N(0,\sigma^2_1),\cdots, x_{k,n}\overset{d}{\...
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1answer
33 views

Question on Rao-Cramer Lower Bound

A question with a solution that I don't quite get: asking for the Cramér-Rao lower bound of a random Poisson sample. If we take the log of the function $f(x; \theta)$ and take its first derivative ...
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1answer
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Consequences of choosing the wrong GLM response distribution

In Chapter 10 of McElreath's Statistical Rethinking (2nd edition), he argues that the response distribution for a GLM should be chosen to maximize entropy given a set of constraints on the response ...
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Equality in distribution - study of a condition

Given a class $\mathcal{P}$ on a Borel measurable space $(\mathbb{R},\mathfrak{F})$. Let $q\in\mathcal{P}$ and $p\in\mathcal{P}$ denote two arbitrary density functions and suppose that the following ...
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Probability of an event happening in a given istant

Simplified context: I'm watching traffic over a computer network. I consider "connection" as an event, which obviously can repeat many times during a period of time. I observe my network for ...
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Ways to sample from a distribution that are more efficient than random

I am trying to sample from a known distribution (somewhat complicated in that a transformed random variable has random noise from a scale mixture of normals added to it and is then back-transformed - ...
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Multivariate rare events

My data is something like this: I have U urns, and I have taken a bag of $n$ objects from each urn. Each urn has $N$ objects, and I have sampled $n$ with replacement. $n$ is comprised of coloured ...
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18 views

Bayesian AB Test Prior Distribution

I am trying to figure out the following issue - while using bayesian method for A/B testing and while comparing control to new variant (i know the control prior data but not the variant), can i use $...
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2answers
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How to transform one Poisson distributed random variable to another with a different mean?

Since the simple affine transformation does not preserve Poisson distribution, I'm wondering if there is any trick to apply a (deterministic) transformation to a Poisson random variable with mean $\...
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If V=X+Y how to prove that there is no independent random variables X,Y make V a uniform distribution [duplicate]

If V=X+Y,how to prove that there is no independent random variables X,Y (with the same distribution function) make V a uniform distribution function on [0,1]
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Finding P value from t (MATLAB)

I am not really familiar with the student’s t distribution. My problem is highlighted in picture, the variables I have obtained are ‘tj’, ‘rj’(partial correlation), N(number of data points) and k(...
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What distribution should be used to model at least 1 “hit” in n trials where the probability of each hit is not constant for each trial?

Obviously the first things that came to mind were binomial/geometric/beta binomial distributions but these all assume probability of hits are constant. As an example to try and illustrate what I am ...
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How do I find the PDF from a multidimensional CDF with indicator functions?

I have what I'm sure is a very stupid question. When I have a two-dimensional random variable $\tilde{X}=(X_1,X_2)$ with the cdf $F(x_1,x_2)=(kx_1^2I_{(0,1)}(x_1)+I_{[1,\infty)}(x_1))(kx_2^2I_{(0,1)}(...
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1answer
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Do conditionals determine joint distribution?

I read some questions on this matter, but I found a tricky example of conditionals, and I don't really know how to approach this problem. Let $X,Y$ be random variables such that $X,Y\in\{0,1,2,...\}$. ...
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1answer
44 views

What are the first steps to find the right model for non-normal data?

I a total of 8 Independent Variables (4 continuous - Scales outcomes - and 4 categorical - Demographic and other personality questions) and 2 Dependent Variables (1 continuous and 1 count). The DVs ...
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1answer
37 views

Density of square root of sum of squared independent uniform random variables [duplicate]

Let $X \sim U(-1, 1)$ and $X \sim U(-1,1)$. We want to find density function of $W = \sqrt{X^2 + Y^2}$. I got stuck and I have no idea, where I am making a mistake. This is my approach. Let $F$ be a ...
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Intuitive explanation of Kolmogorov Smirnov Test

What is the cleanest, easiest way to explain someone the concept of Kolmogorov Smirnov Test? What does it intuitively mean? It's a concept that I have difficulty in articulating - especially when ...
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1answer
39 views

What is the quantity $\delta_x$ at point mass $1$ for any point $x$ in the Influence Function formula?

I'm reading an article on the use of influence curves in robust estimation (Hampel, 1974) which includes the following definition of an influence curve for an estimator $T$: Let $R$ be the real line, ...
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0answers
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Fitting a range of distribution and test for goodness of fit - choose based on p-value or chi-squared values?

I have data for which I want to study the best distribution that fits this data. I am following this blog post to do my experiments. Basically the following things are happening: fit a number of ...
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0answers
28 views

Distribution where mean and mean of reciprocal are both 1 [duplicate]

Is there a non-trivial distribution for a positive r.v. $X$ such that $\mathbb{E}(X) = \mathbb{E}\left(\frac{1}{X}\right) = 1$? If possible, I'd like the distribution to support (have positive ...
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2answers
23 views

Accept-Reject algorithm from binomial or other non-uniform distribution

I'm currently researching monte carlo simulations and the different methods. What I'm finding is that methods such as accept-reject typically sample from a uniform distribution and then compare that ...
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2answers
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Why do I get strange results from the Kolmogorov–Smirnov two sample test?

I'm trying to understand the output of the Kolmogorov-Smirnov two sample test. I have two columns, test1 and test2. I'm having some difficulty in understanding the interpretation of the ...
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1answer
38 views

How to choose a prior : family for a response with negative values?

I’m modeling percentage change in oxygen levels in the blood from a particular experiment. So my prior before seeing the data was an inverse gaussian distribution. But my data (response variable ) has ...
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2answers
27 views

Apply normality assumption on limited sample size

I now have a measurement which only gives me: i) the range of values (continuous in nature), say, 110.0 to 160.0 without the access to the actual underlying data from which these statistics are ...
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1answer
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Interpretation for changes in a $\chi^2$'s density as $k$ increases

The chi-square's density becomes more regular as $k$ increases: $k=1$ unbounded, convex $k=2$ bounded, convex $k=3$ close to 0 near 0, unbounded positive slope $k=4$ close to 0 near 0, bounded ...
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1answer
103 views

Finding the target distribution for the Metropolis algorithm

Let's consider the Markov chain $X_n$ defined on $\mathbf{X} = \{0,1,2...,n \}$, generated according to Metropolis algorithm. Let $X_0 := 0$ be a starting state. The accepting rule is as follows: if $...
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0answers
12 views

Estimating tail deviation from Q-Q plots

I am running experiments and for certain cases am able to find a suitable distribution for the data. However, in most cases, depending on a certain parameter, the observed vs fitted distributions have ...
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0answers
36 views

The quantile of prob=0.05 and mean

What is the meaning of the difference between the quantile of prob=0.05 and mean for a sample form a specific distribution? In other words, I would like to understand the relationship between ...
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1answer
59 views

Is a sum of two binomial distributions with different $p$ also binomial? [duplicate]

I have two independent random variables which follow binomial distributions $X \sim B (n_1, p_1)$ and $Y \sim B (n_2, p_2)$. Can we say that $Z = X + Y$ is also binomially distributed $Z \sim B (n_1+...
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0answers
31 views

Use Central Limit Theorem to determine distribution

Task Let $X_1, X_2, ...$ be a sequence of independent random variables with standard uniform distribution $U(0, 1)$. Lets define $Y_1 = X_1$ and $$ Y_i = \begin{cases} X_i & i \mbox{ is even} \\ ...
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1answer
61 views

How to choose a good test statistic to use in R

I find it hard to understand how to use statistical tests in R. I must use the most compatible statistical test to test if there is a significant difference between the means of 2 samples. I have 2 ...
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2answers
236 views

Is power law distribution for extreme event special like normal distribution?

The power-law distribution is defined as below in Wikipedia article: The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. $$ \mathrm{Pr}[X>x] \sim x^{-...
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1answer
28 views

What is the correct probability distirbution for the given situation?

A game is played until either a player receives their 3rd loss or their 7th win. What is the distribution of the number of wins that the player gets before their 3rd loss? I was thinking of using a ...
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30 views

What is the distribution of sample's range, obtained from a normal distribution?

Let $x$ is a normally distributed variable: $x\sim N(\mu,\sigma$). I wonder, the distribution of the range. I mean, let we have a sample(let size $n$) from the above distribution. Let define $...
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Can I use the square of correlation from copula to calculate the R square (co-efficient of determination)?

I have two variables, security returns, and market returns. I want to calculate R-Squared from copula correlations. What is the possible way to calculate the R-Square from the copula of these two ...
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1answer
33 views

Why does the predictive distribution involve no $\textit{new}$ data?

Given data $(\textbf{X,Y})$, a Gaussian Process $\textbf{F}$ that is normally distributed and weights $\textbf{W}_1$ and $\textbf{b}$ also from a distribution, how come is the following distribution \...
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0answers
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Interpretation of variance of negative log probability

Given some discrete random variable $X$ with pmf $P(X)$ the expectation of its negative log probability $H(X) = E_{X}[-log(P(X))]$ is defined as the entropy which is an important statistical quantity. ...
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1answer
33 views

GLMM with ex-Gaussian distribution function (trial-level reaction time data)

I am trying to use GLMM in R to fit a mixed-effects model (three categorical predictors, one continuous predictor) to trial-level reaction times from a group of participants. The reaction time ...
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0answers
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How can I robustly infer the tail shape of a distribution from data?

I have collected some data, and I want to quantify how heavy the tail of its underlying probability distribution is. I believe that the tail is described reasonably well by a Weibull distribution $W(\...
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1answer
70 views

How to calculate the expected value of k heads in this case?

I'm having some trouble on how to tackle the following problem $X_1$ is a random variable with probability density $f(x)$ in the range $[0,1]$. A value of $X_1$ is picked, call its value $p$. A coin ...
2
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1answer
34 views

Measuring Similarity between 2 Angular Distributions

For a project I'm working on, I have two sets of samples, where each set has N x length 7 vectors. For context, each vector represents the joint parameter setting for a robot (the angle each joint of ...
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0answers
28 views

Orthogonal transformation to derive function of standard normal random variables

$X_1$, $X_2$ and $X_3$ are independent standard normal random variables. Find the distribution of $$T=\frac{X_1+X_2X_3}{\sqrt{1+X_3^2}}$$ Should I use orthogonal transformation to solve the above ...
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To what degree can distribution fitting of a response variable inform GLM family and link selection?

This is more of a theoretical question. I have a response variable that is best described by the Box-Cox Power Exponential distribution, but there is no way to really "run a GLM" with this information ...

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