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A distribution is a mathematical description of probabilities or frequencies.

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$X$ and $Y$ are independent and have the same law then $(X,X+Y)$ and $(Y, X+Y)$ have the same law

My background is in maths so don't have "deep" knowledge or intuition about the topic. The argument that's presented says that can write $(X,Y)=(Y,X)$ so i can write $(X, X+ Y ) = (Y, Y + X ) = ( Y, X+...
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1answer
36 views

How to measure how “good” or accurate a probability distribution is? Entropy, variance or what?

How can one measure the accuracy of the probability distribution of, say, a physical magnitude? I know one good candidate is the entropy, which measures the amount of information one has about the ...
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1answer
21 views

Are there theoretical reasons for choosing between similar distributions?

I'm interested in estimating the distributions of a few skewed datasets, for example extreme heat, and extreme rainfall. There are many distributions that can be fit to these kinds of data, for ...
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1answer
72 views

Distribution of sum of exponentials

Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$. Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-...
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1answer
20 views

Law of total probability - Continuous conditioned on discrete?

Let $Y \sim f_Y (y)$ be a strictly continuous r.v. Let $S \sim p(s)$ be a strictly discrete r.v. Can you write the density $f_Y (y)$ as $$f_Y (y) = \sum_S f(y|S=s)p(s)$$ I know the law of total ...
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1answer
23 views

Question on complete sufficient statistic

In my textbook, I have this example which says $X\sim U(0,\theta)$ we show that the family of PDFs of X is complete. We need to show that $E(g(x))=\int_{0}^{\theta}\frac{1}{\theta}g(x)dx=0\ \ \ \ \ \...
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18 views

Estimation and testing of unconditional distribution in time series models

In time series models, such as ARMA-GARCH, is it possible to estimate what the unconditional distribution is? Given that the time series is auto-correlated / persistent, how many observations would be ...
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3answers
84 views

Conjugate prior, unclear definition

Consider the following definition: A family $\cal F$ of probability distributions on $\Theta$ is said to be conjugate (or closed under sampling) for a likelihood function $f(x|\theta)$ if for every $\...
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1answer
25 views

In a probability generating function, what exactly is the parameter of G(z)?

For instance, given $\DeclareMathOperator{\P}{\mathbb{P}} \DeclareMathOperator{\E}{\mathbb{E}} G(z) = \E z^X$, what exactly is $z$? and also what does the generating function actually give you? ...
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1answer
22 views

How to draw one standard deviation range around the mean of a skewed distribution [closed]

I have a distribution of data with a positive skew, shown in the image. The standard deviation is 1.34 and the mean is 2.01. I want to illustrate on the graph the range of values that are within one ...
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1answer
38 views

probability of repeated events

I have a website and I want to calculate the probability of clicks on the ads. Let the probability that each user clicks on a link be p (something like ...
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16 views

Does p-value (significance) function follows a cumulative distribution function (CDF) for every fixed sample $X$? [closed]

Let's consider a one-sided hypothesis test $H_{0}:\theta \leq b$ vs $H_{1}:\theta > b$, for a given $b$ in the parameter space $\Theta$. Now the p-value function is, P - value function $p_{n} = p_{...
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1answer
14 views

Evaluating usefulness of estimations of a parameter for different distributions

If I had a sample of size n and wished to estimate some parameter, say p for two different distributions from the produced sample what would be required to determine which was more useful? Assume the ...
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0answers
12 views

How to interpret the probability density function exceeding one over a finite interval? [duplicate]

If one looks up 'Frechet Distribution' on wikipedia, one will find the following figure in the top-right of the page I was under the impression that the integral of the PDF function taken from ...
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1answer
26 views

How to fit laplace/exponential distribution to cosine similarities?

I am a computational biologist with little experience fitting data. I'm trying to fit a distribution of cosine similarities computed between sparse matrices. The goal is to be able use this ...
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0answers
39 views

Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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0answers
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Can I use this distribution to model my data according to these plots?

I used the Anderson-Darling and the KS tests to decide whether my data and the distribution fitted on my data has the same distribution. Both tests rejects the null hypothesis. However when I look at ...
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0answers
32 views

Confront Distribution of 2 samples

I found this problem during an interview, and I have little or no clue on how to proceed. Say you have two samples, of different length: ...
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0answers
7 views

How to find parameter $k$ from a negative binomial distribution in R?

I want to find the value of parameter $k$ from my data set. The data set is composed of several populations. Should I calculate the parameter $k$ for each subpopulation, or for the population at large?...
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0answers
5 views

How do I add a standard deviation scaling factor to my cumulative normal distribution function in R? [migrated]

I am using the quickpsy package in R. I would like to have a parameter that scales the standard deviation that quickpsy calculates for the cumulative normal distribution function ...
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32 views

Single Sample vs Population Non-normally distributed

I have rna-seq gene expression data (but could also be reads assigned to species) where most of the genes are expressed at low levels, and the units are discrete ranging from 0 to the total number of ...
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1answer
21 views

How to scale between a equal distribution and an empirical distribution

I am not that good at expressing things mathematically, so I'll start with the practical problem right away: I have a set of four objects: O1, O2, O3, O4. Now I want to assign a variable that scales ...
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0answers
39 views

What do machine learning people mean by $x \sim X$? [closed]

See the question I posted on math stackexchange. https://math.stackexchange.com/questions/2947908/what-does-x-sim-x-mean-in-probability/2947934?noredirect=1#comment6087816_2947934 I just want to ...
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2answers
30 views

Forecasting From an Age-based Distribution

I have an age-based probability distribution that looks something like this, where the age is in rows, and the year is in columns: ...
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0answers
7 views

Loss Function Asymptotic Distribution

I am trying to find the asymptotic distribution of a maximum estimator loss function of the type: $$ \hat{\theta} = \arg \max_\tilde{\theta} M_n(\tilde{\theta},x) $$ $$ \theta = \arg \max_\tilde{\...
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1answer
29 views

How can a dnorm probability be larger than a corresponding cumulated pnorm probability [duplicate]

Using the probability distribution density functions dbinom, dnorm, etc. and the corresponding cumulative probability functions pbinom and pnorm, I noticed that the dnorm density values could be ...
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0answers
13 views

Variance of the average of two (or more) Net Promoter Scores?

As per Wikipedia, "The Net Promoter Score is obtained by asking customers a single question on a 0 to 10 rating scale, where 10 is "extremely likely" and 0 is "not at all likely": "How likely is it ...
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3answers
102 views

what does p( y | μ,σ²) really mean?

Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically: I understand what p( A | B ) where A="I am sick" and ...
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1answer
37 views

Is there a distribution that matches my data?

I am not an experienced statistician, typically only dealing with normally distributed data. I would like to know if the following data sample can be represented by a well-known distribution? If so, ...
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1answer
26 views

Any transformation of Gaussian random variables admits covariance calculation after transformation?

Sigmoid transform of Gaussian random variables does not have an easy calculation of covariance. Specifically, I'm looking for a function transform $$f: \mathbb{R}\rightarrow [0, 1]$$ such that it ...
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28 views

Comparing percentiles of datasets of varying sizes

I have a dataset with 100 observations. I calculate the 25th, 50th and 75th percentile of this dataset. I have another dataset of new observations which contains about 10 values. I calculate the 25th,...
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1answer
28 views

multivariate normal distribution range [duplicate]

Simple question about MVN pdf. I understand the domain to be [0,1]. However, why does scipy.stats.multivariate_normal.pdf output values above this range. E.g. <...
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0answers
19 views

why do we start with FZn(t)=P(Zn<=t) in finding limiting distribution function?

I am learning about convergence now and we did two examples in class. Let $x_1,...x_n$ be iid $U(0, \theta)$, let $Y_n = Max(Y_n)$. Find the limitng distribution of $n(\theta-Y_n)=Z_n$. Let $x_1,......
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1answer
128 views

Is this distribution unimodal, even though there is no data on one side of the mode?

... or does there need to be data on the left side of the mode? The logic to "not unimodal" would be that there must be a peak to be unimodal and there's no peak if the data only decreases on one side....
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Testing if word-count vectors follow a multinomial distribution

I am attempting to make a Naive Bayes classifier for word count vectors (each document is represented as a vector of word counts). For this, I am using SciKit-Learn's MultinomialNB. From what I ...
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23 views

How to obtain cumulative distribution function for a specific probability density function via R

I have the following probability density function. $$f(x,y) = \dfrac{1}{B(a,b,c)}\dfrac{x^{(a-1)}y^{(b-1)}(1-x)^{(b+c-1)}(1-y)^{(a+c-1)}}{(1-xy)^{(a+b+c)}}$$ I wrote the following ...
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1answer
25 views

What can I conclude about the distribution of wrong phone numbers?

Let's say I have a list of 100 phone numbers. I call them all. Nobody picks up for 70. I get someone on the line for 30. Of those, 10 are wrong numbers. What can I conclude about the distribution of ...
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1answer
59 views

Distribution of maximum frequency of uniformly distributed integers

If I roll an M sided dice N times, there will be at least one number that occurs most frequently. What's the distribution of that maximum frequency in terms of M and N? (its pmf and name if it has one)...
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6 views

Nonparametric classification of a sample of values — is my approach correct?

Suppose I have a machine with a number of different labelled settings. The labels go from $1$ up to $L$. When I choose a setting on the dial, let's say setting $j$, I can have it output i.i.d. samples ...
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1answer
30 views

Why random numbers from fitted distribution do not have the same distribution as the sample data?

I have a data set and I would like to fit a t distribution on it. I use R or Python to feed into my data, and I get the degrees of freedom, the location and the scale parameters. After that, I ...
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1answer
19 views

The ways to normalize the likelihood in EM algorithm

In Wikepedia it states that: In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. And ...
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1answer
41 views

Which cumulative distribution of F(X) is equal to the cumulative distribution of its sample median (as sample statistics)

We consider random sampling from a population in which the variable of interest $X$ has some cumulative distribution $F$. Next, we consider a simple random sample of size $n, X_1,\ldots,X_n,$ which ...
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1answer
28 views

How to minimize Chi-Square using the CDF instead of the PDF?

Suppose one has data that is suspected to obey a normal distribution. One computes a histogram of the data, and performs Pearson's Chi-Squared Test. To perform this test, one must compare the observed ...
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21 views

Problem getting SD from a 95% CI of a probability

I'm working on a simulation designed to investigate fertility, birth control, etc. One of the inputs to the simulation is the probability that a given ovulatory cycle is capable of producing a ...
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0answers
8 views

Asserting global labels based on multiple predictions

I’m currently classifying WSI (Whole Slide Images) patches as whether or not they have the label X_label (basically the WSI is 20 000x20 000 large and I make ...
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0answers
26 views

Computing probabilities of a Skellam distribution

Exist a recurrence equation for calculating the probabilities of a Skellam distribution? I need calculate these probabilities in an efficient manner.
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19 views

Kolmogorov distributuon derivation

I would like to know if there is a book talking about the derivation of Kolmogorov distribution (Using usual definition for the bridge process) \begin{align} P(K\leq x)=1-2\sum_{i=1}^{\infty}(-1)^{i-1}...
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17 views

Marginal Likelihood of Multinomial Dirichlet model

To find the marginal likelihood of the multinomial Dirichlet model, I tried the following: $$\int_\theta p(N|\theta)p(\theta)d\theta=\frac{n!}{n_1!...n_K!}\frac{\Gamma(\sum_{k=1}^K\alpha_k)}{\Pi_{k=1}^...
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1answer
68 views

Is there a method to transform all z-scores into positive values?

During my calculations I need to use square roots but z-scores can be negative. Is there a trick to transform them into positive value without missing the usefulness of z-scores? What if I have not ...
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24 views

How can I test if a permutation hypothesis test produces valid results based on the probability distribution used?

I'm using the Join Count statistics to get insight if there is spatial autocorrelation in a categorical feature. I would like to test if the pseudo p-value returned comes from a valid hypothesis test ...