A distribution is a mathematical description of probabilities or frequencies.

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What is the distribution family where one side is with light tail but the other side with heavy tail?

For example the distribution of weights of human. There are not many adults under 40 kg, but a lot more people heavier than 100 kg, although the average of an adult's weight is, let's say, 70 kg. ...
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pdf of a product of two independent Uniform random variables

Let $X$ ~ $U(0,2)$ and $Y$ ~ $U(-10,10)$ be two independent random variables with the given distributions. What is the distribution of $V=XY$? I have tried convolution, knowing that $$h(v) = ...
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Probability distrubution of 3 products

A lightbulb factory produces bulbs in 3 qualities. 50% of production is Low quality, 30% average quality, 20% good quality. A randomly selected lot of 4 bulbs gets tested. Low quality lights have a ...
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Find distribution of Bus arrival time

I am currently working on a problem in my research which can be modeled into the following question: Let's say I have a rich dataset with values for the variable $A$ which is equal to ...
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5answers
284 views

Intuitive explanation of convergence in distribution and convergence in probability

I am quite unsure of the intuitive difference between a random variable converging in probability versus a random variable converging in distribution. I've read numerous definitions and mathematical ...
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Clarify terminology regarding truncated and censored distributions [duplicate]

I'm looking for clarification on the definition of truncated distributions and on terminology for censored distributions and truncated distributions. I recently had a [dialog on SO][1] regarding [a ...
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Developing probability distributions

An event occurs, and after this event has occurred there is a set of conclusions which can be drawn. All of these conclusions have results which are distinct. I am trying to keep this as general as ...
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Why is the p-value distribution not uniform?

While learning on the q-value http://www.totallab.com/products/samespots/support/faq/pq-values.aspx I saw that, under the null-hypothesis, the distribution of the p-value is expected to be uniform. ...
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How to prove brand preference?

Real Case: We have survey data on which of 6 brands is preferred by customers in each of 4 different product usage situations. Our customer segment is well defined and we think we have a ...
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Comparing 5 pt Likert scale answers from two different groups (not the same size) [duplicate]

I have ~150 responses to a number of 5pt Likert scale questions along with lots of demographic data about my respondents. I'm now trying to divide my respondents into groups based on their demographic ...
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30 views

How to estimate mean and standard deviation of a normal distribution from noisy data?

I have $n$ observations, $x_i$ following a normal distribution. I would like to estimate $\mu$ and $\sigma$ from my samples. Normally I would simply estimate $\mu=(\sum x_i)/n$ and $\sigma^2=\sum ...
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rate of convergence of sample mean

Law of large number ensures the convergence of sample mean to population mean. But it does not tell about the rate of convergence. Now CLT tells about the rate of convergence. Then what is the ...
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Defining the probability distribution of a Random vector given the probability over a “sub-vector”

Suppose I want the probability distribution over a random vector $X={X_1 ,X_2 ... X_n }$. What I already have with me is the distribution over a subvector $X_i , X_{i+1}...X_m$, $m<n$ which I ...
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1answer
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Using stat methods on summary statistics

Is it ok to use statistical methods on summary statistics, as they're random variables? I ran into an interesting problem while working with a client, but will try to keep it very general as the ...
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1answer
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Frailty Models: Gamma distributed frailty and Inverse Gaussian distributed frailty

In modelling of frailty using assumptions distributions of frailty are Gamma distributed frailty and Inverse Gaussian distributed frailty. Frailty is unobservable risk factor of mortality. How to ...
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32 views

How to calculate probability that another player has a card

In a poker game, how would we calculate the probability that another player has at least one certain card? Lets say that there are five other player, there are four cards in middle of the table, and ...
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28 views

How is the probability of Weibull Distribution exceed $1$ for some value? [duplicate]

Here is R code which show that probability of Weibull distribution is $1.7$ for some value $t$. But as far i know probability lies between $0$ and $1$. How is it possible ? R code: ...
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marginal conditional distribution from MCMC output

I have a MCMC sampler that targets $$\mathbb{P}(U_1,U_2,...U_n \mid G(U) \leq 0)$$ where $U=(U_1,U_2,...U_n)^T$. I realize now I am more interested in estimating the conditional density $$p_k = p(u_k ...
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Multivariate normal distribution [closed]

Could somebody help me with this problem please? Let X = (X1, X2, X3)′ denote a random vector with distribution N3(μ,Σ),whereμ=(2,1,2)′ and  Σ= 2 1 1 1 3 0. 1 0 1 b) Find the ...
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Variation on the urn problem and frequency distribution

I have $6$ machines each producing different coloured balls. The balls are mixed together in a large vessel. Groups of $6$ balls are extracted at random for packing. Each pack will therefore have a ...
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28 views

KL divergence between a gamma distribution and a lognormal distribution?

Is there a closed-form formula for the KL divergence $D_{KL}(X,Y)$ where $X \sim \mathrm{Gamma}(k,\theta)$ and $Y \sim \mathrm{LogNormal}(\mu,\sigma^2)$ ? Many thanks.
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how is the 2/3 rule effected by a skewed distribution with a zero point? [closed]

If my SD makes one standard deviation below zero and negative, where the data does not go, then does the 2/3 rule stand with 68% between zero and 1 SD and is the missing negative data -1 SD and - 2 SD ...
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35 views

comparing probability histogram

I have two probability histogram samples. I know there are methods(i.e KS test etc) out there to compare histograms but I am trying to compare through simple sum of absolute difference between these ...
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68 views

how can I test if a sample was created from a specific Discrete Distribution

How can I test if a sample was created from a specific discrete distribution. For example, if I have the following distribution 1- 0.2 2- 0.5 3- 0.3 and I ...
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Proof that the probability of one RV being larger than $n-1$ others is $\frac{1}{n}$

This is a follow-on from my previous question about samples from a distribution. Suppose $X_1 \ldots X_{n-1}, X_n$ are random variables all following some fixed distribution $D$. How do I prove that ...
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How to find the distribution of a function of multiple, not necessarily independent, random variables?

If $Y$ is a random variable defined as $Y=g(X_1,X_2)$, where $X_1$ and $X_2$ are two different random variables whose distributions are known (say with pdf's $f_{X_1}$ and $f_{X_2}$), how do we find ...
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When is calculating probability based on a poisson distribution preferred over a binomial distribution?

I'm trying to understand why a poisson distribution may be preferred over a binomial one when modeling binary cases. Is there a case where you can't use a binomial distribution to solve a problem ...
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To BIN or not to BIN a continuous data to get a Fragment Size Distribution ?

I have a data set in excel of almost 6000 entries (quantitative and continuous => P(X=x)=0, I mean the possible values for my continuous random variable X are uncountably many). Each point will ...
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46 views

Get probability distribution function from density function

For a given density function, how does one find its distribution function? For example, I have a density function: $f(x)= \begin{cases} t ^2 / 9 & \text{if } t \in (0,3)\\ 0 ...
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Data Sampling while preserving the underlying distribution

I have a large 10-15 dimensional data set with close to 10 million points. I want to test some algorithms over a chunk of this data. But I don't want the character of this data to be lost by selecting ...
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what is the name of distribution similar to von mises distribution

I am wondering whether the following PDF is a named distribution. Over the interval $0< \theta < \pi$, the PDF of the distribution is written as $$ P(\theta) = \frac{2^{-n/2} k^{n/2} e^{k \cos ...
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Interpretation of cartesian product of the support of marginal distribution

Suppose we have a multivariate data set, $s = (s_1, s_2, ... s_p)$ and each $s_i$ is distributed with a distribution that has finite support (we'll call each $s_i$ a "source"). Let us denote the ...
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Finding a distribution for data in $\mathbb{N}_0$

Suppose, we have a set of 10,000 individuals. Each individual falls into exactly one of 200 categories. [Edit: The categories are phenotypes (different potential outcomes) of the one property that is ...
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25 views

Question about posterior mean calibration

I'm reading the article "Prior distributions for variance parameters in hierarchical models" by Andrew Gelman(link). This is an extract that I don't understand very well: Posterior inferences can ...
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Generating a skewed distribution given the median and left and right “$1\sigma$” limits [duplicate]

Edit: I found a solution to the problem, which is at the bottom of the post. I'm going to leave the post as it is in case someone else encounters a similar problem! I was banging my head to a ...
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Doubt about conditional conjugate priors

I've just read the definition of conditional conjugate prior in this discussion but I have still some doubts. According to the definition given, it seems that the prior distribution of $\theta$, ...
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1answer
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Multivariate t-distribution definition

I have the following marginal posterior of a vector $\phi$ ($p$ by $1$): $$p(\phi | Y) \propto \left[1+\frac{1}{h}\left(\phi - \tilde{\phi} \right)' \Gamma \left(\phi - \tilde{\phi} \right) ...
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What's the Mode of a Bivariate Poisson Distribution?

I have been looking at the bivariate Poisson distribution and I am wondering if there is close form expression for the mode of this distribution. I know the mode of the univariate Poisson distribution ...
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Q-Q plot and sample size

To study the quantile-quantile plot, I used the following codes (modified from here). The first group of pictures is derived from 100 data points while the second from 10000. ...
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Test for unimodal symmetric distribution

I am trying to test whether data from a sample I have follows a t distribution with n degrees of freedom for a given n. I am looking for something more powerful/recent than Kolmogorov-Smirnov test. ...
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302 views

How to detect random clicks on a page?

I have been tasked with detecting if a bot is clicking randomly around a web page. I was thinking of splitting up the web page into 20 by 20 pixel squares, and then assigning each click to a square ...
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Is calculating a percentile the same as evaluating a cumulative density function?

I'm trying to make the jump from the idea of a percentile, say, over the real number line (where the nth percentile is simply the position in which n% of data points are below it, and 100-n% are above ...
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The probability of a random variable being larger than a sequence of random values

Suppose we have a fixed, known, $n$, and each $x_1 \ldots x_n$ is a random number generated uniformly over $[0,1]$. What is the probability that $x_n$ is the largest value in the sequence?
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How to calculate the likelihood function

Lifetime of 3 electronic components are $X_{1} = 3, X_{2} = 1.5,$ and $X_{3} = 2.1$. THe random variables had been modeled as a random sample of size 3 from the exponential distribution with parameter ...
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On using one distribution to approximate another

Every basic text about different statistical distributions make notions about when one distribution can be approximated using another one. For example, we are told that the binomial distribution with ...
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Deriving the marginal posterior

Context of the question: You can take everything below as given. $E_2$ is a $k$ by $1$ matrix and $V_{22}$ is a $k$ by $k$ matrix. Let $X$ denote the data. I have derived so far the joint posterior ...
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How to find $\displaystyle \dfrac{d}{dt} \left [\int_t^\infty xf(x)~dx \right ] $ (when $f(x)$ is a probability density function)

How can I solve this? I need intermediate equations. Maybe the answer is $-tf(x)$. $\displaystyle \dfrac{d}{dt} \left [\int_t^\infty xf(x)~dx \right ] $ $f(x)$ is probability density function. That ...
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When is an ellipsoidally (elliptical) distributed random variable spherically symmetric?

If $Y$ is ellipsoidally distributed, and $\mu_y \propto 1_p$ and $\sum_y = \sigma^2 I$ is $Y$ spherically symmetric? EDIT: added - Is it exchangeably distributed? Please give a proof not a yes/no.
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Exponential family distribution with high-order statistics

An exponential family distribution in its simplest form is given by $p(x|\theta) = \exp(\theta^\top T(x) - A(\theta))$ where $T(x)$ is a vector of sufficient statistics, $\theta$ is its natural ...