A distribution is a mathematical description of probabilities or frequencies.

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Using influence curves/functions to find limiting joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
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Applying Delta Method to generate prediction bands for custom distributions step by step

I think that this is an interesting issue for all those who whan to not only adjust one prediction function, but also to obtain the prediction bands to asses the graphical or tabular error I'm trying ...
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40 views

$E(X_i \cdot I(X_j>\theta))$, where theta is the median?

Let $X_1, ..., X_n$ be iid with a distribution F. Let $\theta$ be the median of F. What is the value of $E(X_i \cdot I(X_j>\theta))$? If $i\neq j$, then $E(X_i \cdot I(X_j>\theta))= 1/2 \cdot ...
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164 views

Test that two normal distributions have same standard deviation

I have two normal distributions, and I want to test whether they have the same standard deviation, I really don't care about the mean. My idea is: de-mean both of them and then use Kolmogorov-Smirnov ...
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18 views

Compare skewness of many distributions with few observations

I have a dataset with page view data for about 500,000 users, divided into two groups. Each user can visit up to 5 pages, each as many or as few times as they want. So for each user, I have the ...
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44 views

Distance between two random vectors

I have two random vectors, $A$ and $B$ with each consisting of $n$ geographical co-ordinates $(x_1,y_1),(x_2,y_2)\dots (x_n,y_n)$ and $(\tilde{x}_1,\tilde{y}_1),(\tilde{x}_2,\tilde{y}_2)\dots ...
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72 views

Which distribution for modelling duration of tasks?

Recently I was present with a task estimation technique. Instead of letting people rate a task for x - amount of hours, I let them discretize tasks into discrete sizes like small / medium / large / ...
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27 views

Convergence of Gini index

Let $\theta(F)=2\int^1_0(t-q_F(t))dt$, where $\displaystyle q_F(t)=\frac{\int_0^tF^{-1}(s)ds}{\int_0^1F^{-1}(s)ds}$. For discrete distributions I'm assuming that $F^{-1}(s)=\inf\{x:F(x)=s\}$ (the ...
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55 views

If $X \sim$ unif$(0,1)$, what is the distribution of $U=\max \{X,1-X \}$?

Let $X$ be uniformly distributed in $(0,1)$ and set $U=\max \{X, 1-X\}$. How can I find the distributiopn of $U$? My first thought was to consider this a mixture distribution problem and use the CDF ...
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1answer
18 views

Is there a significantly different number of people between groups?

How do I estimate whether a given number of people across groups (say, 72 people in group 1, 95 in group 2 and 70 in group 3) significantly differs from a random distribution of people across groups? ...
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29 views

Normal Bayesian Model: Marginal distribution of $\bar X$ with unknown mean and unknown variance

For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i, \sigma^2 & \stackrel{_\text{ind}}{\sim} N(\mu_i, \sigma_i^2) \\ \mu_i & ...
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54 views

Sum of lognormal distributions

I am to find the expected value, variance and, preferably, the distribution of the variable $z_n$, where $z_n$ is given by \begin{equation} z_n = \exp\left\{ \Delta t \sum_{i = 1}^n k_i ...
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24 views

Distribution of random vector divided by it's norm

Does anyone know anything about the distribution of a random vector divided by it's norm? More specifically, if we know the distribution of a vector valued random variable $x$, what can we say about ...
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53 views

Seeking a continuous, parametric, bimodal sampling distribution for proportions

I am seeking a parametric probability model whose pdf has the following characteristics: (1) it is supported on a variate axis that is bounded between 0 and 1; (2) it is continuous; and (3) it is ...
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23 views

How to test whether a distribution follows a power law?

I have the data of how many users post how many questions. For example, [UserCount, QuestionCount] [2, 100] [9, 10] [3, 80] ... ... it means each of the 2 users posts 100 ...
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22 views

X vs x Notation in law of large numbers

This may be a silly question, but I can't find a concise answer. I've been studying Convergence of Random Variables in Wasserman's All of Statistics, which starts out by explaining: $X_n$ is a ...
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152 views

Deriving the bivariate Poisson distribution

I've recently encountered the bivariate Poisson distribution, but I'm a little confused as to how it can be derived. The distribution is given by: $P(X = x, Y = y) = ...
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9 views

Comparing file sizes day on day for outliers, what test to use?

I am quite new to statistical analysis, and have to crunch through several thousand folders/files each day to, at a first glance, see if the file size indicates an error (too large/small etc). The ...
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119 views

Question about Dynkin Lehmann Scheffe Theorem

I'm self-studying for an examination, and I would like to understand how to use the Dynkin Lehmann Scheffe theorem for an applied question. I am using Bickel and Doksum's "Mathematical Statistics" ...
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26 views

Estimating a distribution from a dataset with multiple parameters

How would you go about solving the following problem? You're an insurance company who writes workers compensation policies. You want to build a probability distribution for the number of annual ...
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29 views

How to separate distributions from weighted dataset?

I’m trying to separate two component distributions of an apparent finite mixture from a weighted dataset (determined by a weighted.histogram). I've a set of data and weights only for a part of the ...
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39 views

How to normalize bimodal (or multimodal) distributions?

If I have multiple data series, a = [a1, a2, ... a100] ~ bimodal with mu_a1, mu_a2, sigma_a1, sigma_a2, b = [b1, b2, ... b100] ~ bimodal with mu_b1, mu_b2, sigma_b1, sigma_b2, c = [c1, c2, ... ...
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70 views

Algorithm for rating books: Relative perception

So I am developing this application for rating books (think like IMDB for books) using relational database. Problem statement : Let's say book "A" deserves 8.5 in absolute sense. In case if A is ...
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Efficient scale & shape parameter estimation for generalized secant hyperbolic distribution needed

the (symmetric) generalized secant hyperbolic distribution GSHD is very flexible but I found not much at all on how to estimate its 3 parameters. Given the location, I need to obtain scale & shape ...
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90 views

Distribution function of maximum of n iid standard uniform random variables where n is poisson distributed

I am studying probability theory on my own and am trying to work the following problem in the book - Let $X_1, X_2, . . .$ be independent, $U(0, 1)$-distributed random variables, and let $Nm \in ...
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23 views

Approximation by uniform distribution

Imagine I have a random variable $X$ drawn from the sample space $S$ such that $Prob[X = x] = p_x$. Let $A \subseteq S$ be a subset of values from the sample space. My question is if we have a lower ...
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1answer
49 views

Derive the mean of a discrete probability distribution

I am reading inference statistics by casella and berger. They are deriving the general formula for the probability distribution like that: I do not get the step before the result. I would ...
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12 views

Tail index using hill estimator in R [migrated]

As part of my data analysis (of heavy-tailed data) I wish to calculate the tail (for both left and right) indices of around 100 time series of company returns. My data is stored in a large zoo object, ...
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35 views

Multivariate Hypergeometric Distribution with “or more” Successes

I understand how to calculate multivariate hypergeometric distributions. For example, if you have an urn with 2 red marbles, 4 white marbles, 8 blue marbles, and 12 orange marbles, the probability of ...
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Standardising data within-subjects when analysing distributions

Analysing behavioural data, Spivey, Grosjean and Knoblich, (2005) wanted to show that their results came from a unimodal distribution, rather than from averaging over subpopulations in a bimodal ...
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26 views

Which distribution to parametrize price histogram with exponential tail?

I have plotted price histograms which seem to follow and exponential tail very well. However, I would also like to parametrize the initial peak in the low price range. Can suggest what ...
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26 views

What is the distribution of inverse Wishart times a Fisher-Von Mises?

Suppose $p$-dimensional vector $x$ takes a Fisher-Von Mises distribution with parameters $\kappa, \mu$, and $M$ is inverse-Wishart with $n$ degrees of freedom and matrix parameter $I_p$, the $p\times ...
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66 views

Unimodal or bimodal data (MATLAB)?

I am trying to figure out what I did wrong or what I could do to get accurate results. I have n vectors of data, and I am trying to decide whether each dataset is unimodal or bimodal. I assumed that ...
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22 views

Range of successes for multivariate hypergeometric distribution [duplicate]

I'm trying to calculate the probability of drawing specific cards in a deck. I know that I can use multivariate hypergeometric distribution to calculate for a defined success. I'm wondering how I ...
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1answer
27 views

How can I determine if two features, given a set of data, are the same feature or not?

Thanks in advance for the help. Suppose I have a set of data of the form [feature, rating]. For the sake of argument lets assume feature can be movie A or movie B and rating is a ranked, continuous ...
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1answer
64 views

Posterior distribution of a random variable

Im not understanding the following; suppose $y \sim N (\mu,\sigma^2)$ and we have a prior $\mu \sim N (\mu_0, \sigma^2_1)$ Then we can figure out the posterior distribution. What i dont understand ...
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51 views

Moment generating function if the PDF is $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$

Let $z$ a random variable with PDF : $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$, where $0\leq z \leq d$, $F$ is the Hypergeometric function, $k$ is a positive integer, $-a+k+1 ...
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26 views

Stratified random sampling and its distribution

I'm learning stratified random sampling and I'm having hard time to solve this problem. consider the following population of six units: ...
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44 views

Is it possible to build a curve with a given skewness and kurtosis?

I have data in the following form: Day Kurtosis Skewness Total Inflows 163 .3 .5 435670 I'm attempting to do some modeling of an inflow ...
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26 views

Radial distribution to cartesian coordinates

I'm trying to go from a radial distribution in polar coordinates to cartesian coordinates. I'm using Matlab and have tried pol2cart, but are having some trouble getting a matrix with probabilities. ...
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32 views

Conditional or Joint Probability under Various distributions

In various statistical models the baseline equation (like in Naive Bayes $$\mathrm{classify}(f_1,\dots,f_n) = \underset{c}{\operatorname{argmax}} \ p(C=c) \displaystyle\prod_{i=1}^n p(F_i=f_i\vert ...
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19 views

Compare one distribution with multiple other distribution

I've a distribution of numeric scores computed using real data ( my reference distribution ). After that I generated 100000 random distributions of numeric scores computed using points picked at ...
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42 views

Variable bounds from infinity to negative infinity - Change Limits

I have a variable whose values vary between $-\infty$ and $\infty$. I'd like to change its limits from $0$ to $1$. I want to give extremely low values when the values are less than $-1$ or greater ...
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1answer
34 views

How to combine distributions

If I have two continuous distributions $f(x)$ and $g(x)$, there are several mathematical ways to combine $f$ and $g$ to get new distributions. Which correspond to what statistical interpretation? For ...
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1answer
79 views

How to calculate the scale parameter of a Cauchy random variable

Let $(X_n)$ be iid random variables and suppose they have mean 0 and follow Cauchy distribution. I know I can set the location parameter to 0. My question is how to find the corresponding scale ...
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1answer
54 views

What transformations preserve the von Mises distribution?

The von Mises distribution is entirely defined on the circle with a density given by $$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$ where $x$ is in an arbitrary real interval of ...
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47 views

What are the chances of something *not* occurring?

Suppose I'm gambling using a strategy of doubling my bet whenever I lose to recoup my losses from previous bets. If my initial bet is 1/2048 of my capital, I can bet 10 times before I run out of ...
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58 views

How to rigorously test if variables are drawn from a certain distribution

Say I have a list of numbers $X = \{x_1, x_2, \dots, x_n\}$, and I expect them to be drawn from a certain distribution. For my case it is the Binomial distribution $P(x) = \binom{n}{x}p^x(1-p)^{n-x}$, ...