Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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44 views

Distribution of $Y$ when, $Z = Y \times X$, $X \sim Poisson(\lambda_x)$ and $Z \sim Poisson(\lambda_z)$

Is there any know distribution such that when a random variable from it is multiplied with an IID random variable from a Poisson distribution, it results in a Poisson distributed random variable? ...
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I am confused about the sum indices of these posterior distribution formulas

I am reading this notes and trying to understand how the posterior distribution formulas of the variational variables involved have been calculated. I am confused about the indices of the summation. ...
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37 views

How to compute the PDF of a conditional distribution [duplicate]

Let $T \sim Unif(0, 1)$. Then, $f_T(t) = 1 \text{ for } t \text{ in [0, 1] (0 elsewhere)}$. How do we formally compute $f_{T \mid T > 0.5}$? Intuitively, $f_{T \mid T > 0.5}(t) = 2 \text{ for } ...
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Can I use the ROC curve to compare two distributions?

The ROC curve for two distributions $F$ and $G$ can be defined as $$\mbox{ROC}(u) = {F}(G^{-1}(u)),$$ for $u \in (0,1)$. So, if $F=G$, then $\mbox{ROC}(u) = u$. Can I use this property to compare the ...
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2answers
100 views

'Aspect ratio' as a measure of the 'tall-and-skinny' property of unimodal distributions

Kurtosis has not entirely lived up to being a measure of the 'tall-and-skinny' property. If you can access it, see Westfall 2014 for a valuable discussion of the pitfalls of using kurtosis as '...
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35 views

Simulate drawing marbles from a bag with replacement time efficiently

I have a bag with 256 marbles, each a different color. Everytime I run the experiment, I have a 1/16 chance of drawing any marble. I can simulate this by instead considering a bag with 16×256 marbles, ...
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11 views

Determine which independent variables are relevant by examining probability distributions?

I am looking to simulate patient outcomes from a data set that contains patient data such as age, diagnosis, some clinical variables, survival time after treatment and so on. My plan was to use a ...
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1answer
26 views

What happens to the data distribution and results if we calculate z-score of a z-scored data?

The data that I am using is already z-scored and batch normalized. I accidentally calculated the z-score again and then performed further analysis and calculated results. Does it make sense to take z-...
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35 views

Euclidean Norm normalized Normal Distribution

Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anistropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different. ...
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Example distribution where 74% of probability is above the mean

Watching Why You Should Want Driverless Cars On Roads Now, at 8:14 Derek Muller claims: Surveys show 74 % of people believe they are above average drivers. This claim motivates my question, but some ...
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22 views

Understanding natural parameterization of exponential family

I'm going through section 3.4 on exponential families in Statistical Inference by Casella and Berger. They first cite the following general form of an exponential family: $$f(x|\mathbf{\theta})=h(x)c(\...
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23 views

Preconditioning

Problem Statement Let $\pi(x)$ be my target distribution and suppose $\text{Var}_\pi[X] = \Sigma$. Suppose also that I have obtained samples $X^{1:N}\sim \pi$, computed their sample covariance matrix $...
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1answer
30 views

Intuition for hypergeometric variance?

I'm trying to learn the major facts about a bunch of probability distributions, hypergeometric included. I can use the commonalities between it and a binomial to my advantage for thinking through some ...
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0answers
29 views

Why use the Wald statistic for Barnard's test?

Suppose we have a $2\times 2$ table of observations $X_0=(a_0,b_0,c_0,d_0)$ with given sample sizes $n_1=a_0+c_0$ and $n_2=b_0+d_0$. We want to test the hypothesis that the probabilities of success in ...
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10 views

Are highly skewed dependent variables more susceptible to overfitting?

I'm developing a linear regression model to predict wildfire sizes. Here the distribution plot for my wildfire data: I determined my model was overfitting by comparing the R2 with my training data to ...
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1answer
70 views

Conditional probability of sum of $n$ Poisson variables given sum of subset of $n$ variables $P(X_1+X_2+...+X_n=z|X_1+X_3=y)$

I have $n$ Poisson random variables $X_i$ each sampled from an independent Poisson distribution parametrized by $\lambda_i$. I want to compute $P(X_1+X_2+...+X_n=z|X_1+X_3=y)$. The distribution of the ...
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0answers
44 views

Approximating the Logit-Normal by Dirichlet

There is a known approximation of the Dirichlet Distribution by a Logit-Normal, as presented in wikipedia. However, I am interested in the reverse, can I approximate a logit-normal by a Dirichlet? I.e....
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1answer
58 views

Kullback-Leibler distance calculation for discrete distributions?

I have the following model $$N \sim Pois(\lambda) \\ n \sim Bin(N,p)$$ for which I calculate the posterior for the parameter $N$ as $$\pi(N|n,p,\lambda) = \frac{f(n|N,p)\pi(N|\lambda)}{f(n|p,\lambda)}...
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1answer
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Newbie: Difference between the meaning for the data analysis: proportion of correct responses and probability of correct responses

Very newbie, sorry if the question seems to be not very smart... If proportion of correct respones is defined as Number of hits + Number of correct rejections / Number of trials Is the probability of ...
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Would Performing a T-Test between Nonequal Continuous Distributions Provide Valid Results?

I have two samples, and in performing a Kolmogorov–Smirnov test the sample distributions are shown to be significantly different from each other. In knowing that the two sample distributions are ...
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1answer
65 views

Meaning of NOT having a distribution on data [duplicate]

In statistics and machine learning, a common starting point is to assume some unknown distribution $\mathbb{P}$ on the cartesian product $\mathcal{X} \times \mathcal{Y}$ of input space and output ...
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1answer
64 views

What is area under cumulative distribution represent? [duplicate]

Exponential distribution has following probability density function which explains the curvature of a line (For simplicity I am just going to work with x>=0): f(x) = lambda e^{-lambda*x} to find ...
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1answer
37 views

Using PDF values for Likelihood [duplicate]

Given that PDF value $𝑓_𝑋(𝑥)$ for a particular $𝑥=𝑥_1$ does not have any probabilistic meaning (by definition $𝑝(𝑥=𝑥_1)=0$). We still see the use of $𝑓_𝑋(𝑥_1)$ as its likelihood. My ...
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Using Sigmoid in Maximum Likelihood Estimates [duplicate]

I have two questions regarding the use of Sigmoid in MLE: Clearly, the Sigmoid Function is not a PDF. But in the MLE of Logistic Regression, we see Sigmoid being used as if it is a PDF. Is my ...
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1answer
15 views

Fitting distribution to literature-reported incidence rates or cumulative risks

I am building a simulation model in R. In this model, I would like to simulate a patient with a certain baseline age, and then simulate the time-to-event for breast cancer. I want to sample the time-...
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1answer
96 views

Expectation and distribution of ratio of correlated Gamma/Chi-square random variables

This question is very similar to: Distribution of the ratio of dependent chi-square random variables But the big difference is what happens when we don't have standard normal variables. I want to ...
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Confidence Interval using % changes (Trying to classify whether variation from SMA is significant)

0 So I'm attempting to figure out when a drop in the price of a security is "statistically significant". The distribution of the difference is log-normal (makes sense because it can only be ...
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1answer
43 views

Finding a,b parameters if Highest Posterior Density is known

I know that a beta distribution with unknown parameters a,b has a 95% HPD of [0.25, 0.75]. What is the correct approach to solve for a,b?
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Why aren't Normalizing Flows suitable for Discrete Distributions?

I am currently trying to understand why normalizing flows are not applicable to discrete distributions (a quick primer on NF can be found here). The assumptions on the transformation f between the ...
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28 views

Trying to find a distribution for time dependent drought

I would like to seek your support on a modelling issue for which I could not find relevant past postings or published literature resolve it. I am running a cost benefit model to assess the impact of ...
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1answer
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Bayes theorem with multiple draws

Setting I have a question on the "Cookie Problem Revisited" exercise from Allen Downey's Think Bayes 2e. The Bayes theorem is defined as: $$ P(H | E) = \frac{P(H) \ P(E | H)}{P(E)} $$ where ...
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How to calculate alpha and beta parameters from an known mean and variance in normal-inverse gamma distribution

How can I calculate the $\alpha$ and $\beta$ parameters for a normal-inverse gamma distribution if I know the mean and variance?
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1answer
37 views

Exponential family admissibility of base measure, sufficient statistic and log partition function

Let $$ f(y | \eta) = h(y) \exp\left( \eta^\top T(y) + A(\eta) \right)$$ be the exponential family with base density/pmf $h$, sufficient statistic $T$, log partition function $A$ and natural parameter $...
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2answers
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True mean of a truncated distribution?

I use C++ GSL library to generate random numbers now. The numbers obey a distribution, (e.g. normal or lognormal distribution). This library requires the input of expected value ${\mu}$ (i.e. mean) ...
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50 views

Fitting distributions to censored and uncensored data in R

I need to fit lognormal, Pareto, and generalized Pareto distributions to some empirical data that is a combination of censored and uncensored data. I tried using the function ...
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ISI PCB NC$9$ Limiting Distribution of Bernoulli to Poisson

Let $X_i\sim (i.i.d.)$, Bernoulli($\frac{\lambda}{n}$), $n\ge \lambda\ge 0$. $Y_i\sim (i.i.d.)$, Poisson($\frac{\lambda}{n}$). $\{X_i\}$ and $\{Y_i\}$ are independent. Define $T_n=\sum_{i=1}^{n^2}X_i$...
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Can probability distribution predict the electricity consumption of the coming months? if no what do you recommend?

Can probability distribution predict the electricity consumption of the coming months based on a historical consumption data? can it give me a figure? If NO or not recommended for such case, what do ...
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1answer
36 views

How to generate data from a generalized Dirichlet distribution?

I need to generate data from a generalized Dirichlet distribution in Python to test my model, but I have no idea how can I proceed with that, can anyone guide me?
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1answer
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What is the importance of non-informative prior in Bayesian Inference? [duplicate]

By the name, noninformative prior, the prior distribution doesn't contain any information about the parameter. Then why we use this thing to estimate the parameter by the Bayesian approach?
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Deriving the Chi Square distribution [duplicate]

Let $X_1,...,X_n$ be IID random variable such that $X_i \sim N(0,1)$ for all $i=1,...,n$. Derive the distribution (i.e. give the pdf/cdf) of $\sum_{i=1}^{n} X_i^2$. I know that from the definition of ...
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1answer
37 views

Distribution of sum of possibly non-independent Bernoulli random variables with known variance-covariance matrix

I wonder if there are any results concerning the distribution of sums of possibly non-IID Bernoulli random variables when covariances in all pairs of r.v.'s are known. To make this more concrete ...
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2answers
127 views

How to sample from truncated distributions using scipy?

I have min and max values for certain variables such as: Expenses Loss Growth I'd like to add a distribution around them and plot a histogram in python. Distribution could be beta, gamma right type ...
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Using convolution formula

I would like your help to solve the following exercise. I have also reported below my attempt, but I'm not sure I'm properly manipulating the various integrals involved (in particular the switching ...
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2answers
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How to know which set is most likely to have been generated by some probability distribution?

I have 3 sets of points in $R^n$: $X$, $A$ and $B$, how can I (preferably quickly) check which of the two sets $A$ and $B$ is most likely to have been generated by the same probability distribution as ...
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27 views

Power Generalized Weibull (PGW) Distribution

I'm trying to understand Power Generalized Weibull (PGW) distribution for my final project. PGW distribution is the generalization of Weibull distribution. The cdf of PGW distribution is $G(x) = 1 - e^...
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1answer
31 views

Exponentiated Weibull-logarithmic Distribution

I'm trying to deduce the marginal cdf of $Y$ in Exponentiated Weibull-logarithmic Distribution from this paper: Exponentiated Weibull-logarithmic Distribution: Model, Properties and Applications In ...
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64 views

Find the conditional distribution to calculate a probability

I have a question that has been posed to me with 4 multiple choice answers. I cannot see how any of these answers are correct. Can somebody please let me know how to solve this? I'm getting $e^{-1}$ ...
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Bayesian linear regression with given distributions X, y instead of pairs {(X1, y1),..(X100, y100)}

I'm wondering if is it possible to model data by knowing only distribution of features (X) and targets (y). Thus, instead of paired variables {(X1, y1), (X2, y2), .., (Xn, yn)} I know only mean value ...
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5 views

Age-adjusted mortality rate given age distribution and total number of deaths

I am a bit confused as to how I can calculate age adjusted mortality rate given the two datasets that I have. My first dataset consists of covid deaths per city district and is a gross sum (...
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How do you report error bars / uncertainties for a set of probabilities obtained from a hidden Markov state model?

I have a system I want to model with a hidden Markov state model. It has almost 30 different possible observations so I was doing some tests on a simpler system to get a handle of things. Preliminary ...