Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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Canonical form and exponential family

Suppose you have a random variable X, who's distribution depends on $\theta$. If X is a part of the exponential family of distributions, X can be written in a certain form, namely: $$f_\theta(x)=h(x)*...
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What is the limiting distribution of $\chi_r^2$ random variable, where $r\to 0^+$

What is the limiting distribution of $\chi_r^2$(Chi-square) random variable, where $r\to 0^+$. The following picture shows that as $r\to 0^+$ the distribution become degenerated in zero point. If it ...
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How to find the type of my distribution?

I checked the normality of my data on SPSS and one of the variables is not normally distributed. I have the mean, standard deviation, skewness, kurtosis , min and max values of my distribution. But I ...
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Additivity property of Poisson Distribution for non-random variables?

I know that the Poisson distribution is additive, i.e., $X \sim Po(\lambda)$ and $Y \sim Po(\mu)$, then $X + Y$ has $Po(\lambda + \mu)$. I am currently reading a paper that proposes a statistical ...
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distribution for the expected value of the mean [closed]

May I know if there is a distribution for the expected value of the mean of beta variables?
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Prove that $E[\log(\alpha X_t^2)] < 0 $ implies $\alpha < 3.5622$ with $X_t \sim N(0,1)$

I am trying to prove this statement: If $X_t \sim N(0,1)$ then $$E[\log(\alpha X_t^2)] < 0 \implies \alpha < 3.5622$$ which is a a necessary condition often found in textbooks for strict ...
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Probability of difference for uniform sample of consecutive numbers [duplicate]

Lets assume I have a set N which consists of all natural numbers from 1 to n. Now I take a uniform sample of size S from this set. Intuitively I wonder how large the probability is to have a gap ...
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flexsurvreg - shape and scale proportional hazard model

Please see below codes and plot for fitted curves to Kaplan-Meier data (weibull in this example) using the ovarian data. Based on the estimates, the shape and scale of the fitted Weibull distribution ...
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Estimating the distribution of the maximum of N values drawn from N different normal distributions

Let's say we have N distributions $\mathcal N(\mu_i, \sigma_i)$, each with known mean $\mu_i$ and known standard deviation $\sigma_i$, $i=0,...,N-1$. For each $i$, 1 random samples is drawn from from ...
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Interactive plot of Region in Economic Model

So let's say I have the following framework. Let $s_1 \sim N(\mu_1,\sigma_1)$ and $s_2 \sim N(\mu_2,\sigma_2)$. Denote by $f_i$ and $F_i$ their pdf and pdf respectively. Define the following two ...
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A Sample is a a Single Data Point, or a Pool of Data Points?

This question has confused me a lot in statistics. I think in Statistics, a sample is a pool of data points from the PDF, rather than a single data point, am I correct? In everyday language if you ...
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When I repeat an experiment 10 times, do I have 10 different random variables all of which are from that sample space?

I toss 2 dice to get their sum; now, this means I have a sample space from 2 to 12. Now, when I repeat this experiment 10 times, do I have 10 different random variables all of which are from that ...
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How do I calculate the probability density function for a joint beta + uniform distribution?

The PDF of a Beta distribution is $$ f_X(x) = {{x^{a-1}(1-x)^{b-1}}\over {B(a,b)}} $$ and CDF $$ F_X(x) = I_x(a,b) $$ The PDF of a uniform distribution is $$ f_Y(y) = {1 \over {b-a}} \space for \...
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Understanding Dirichlet Distribution Variance

I need some help in understanding the variance/standard deviation in the Dirichlet distribution. I apologize in advance for the lack of latex. In the Beta distribution, as the shape parameters ...
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Does the random variable “Number of Years” follow a Poisson probability distribution?

I am trying to figure out whether the random variable "Number of Years (since an event)" follows a common probability distribution. Specifically, I am tempted to say that it is a Poisson ...
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How to Solve For the Inverse Cumulative Distribution Function of a Double-Exponential Probability Density Function

I'm stuck on figuring out how to sample data from a fake/known double-exponential PDF for a lab project involving C. elegans egg-laying rate data. I need help with figuring out if there's an exact ...
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How to compare count distribution of multiple categories across months and years

I have animals counts organized into six size classes. These were taken across six months in three different years. I would like to compare the distribution of counts across size classes, between ...
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Bootstrapping variance in R gives weird shaped distribution- how to obtain confidence intervals?

this is the first time I've used bootstrapping so it's quite basic! I'm trying to obtain confidence intervals for the standardised variance- defined as the variance over the square of the mean- across ...
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What family of full support probability distributions satisfy that the density of any point in the domain vanishes as the variance goes to infinity?

Let $f(x,\sigma^2)$ be a representative element of a family of PDF's with full support over the reals that is indexed by their variance $\sigma^2$. Under what general conditions of the family of ...
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X is a binomial random variable, then is aX+b follows a binomial distribution? [duplicate]

I know when X ~ N(mu, sigma^2), then aX+b follows normal distribution. But I'm curious when it happens to binomial random variable. And I don't intuitively understand how multiply constant to binomial ...
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Peaks of estimated probability distribution are always lower than those of true distribution - why?

I have some code (shown below) for sampling from and then estimating a normal mixture distribution in one dimension. When I plot the estimate (blue) against the true distribution (black) I get images ...
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Linear transformation of a binomial random variable?

Let $X$ be a binomial random variable with parameters $n$ and $p$. Next, let: $$ Y = aX + b $$ I know that: $$ \mathbb{E}[Y] = \mathbb{E}[aX+b] = a\mathbb{E}[X] + b = anp + b \\ \text{Var}(Y) = \text{...
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Reference Request regarding a sampling problem?

I have a function $f:\{0,1\}^n \rightarrow \{0,n^{L}\}$ that maps each $x \in \{0,1\}^n$ to a class named from $0$ to $n^{L}$, i.e there is an exponential sized population and a polynomially many ...
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Measuring the overlap between two probability distribution [duplicate]

I have many probability distributions, I need to compute the amount of overlap between two probability distributions. I don't know the type of distribution since it really depends on the data itself. ...
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How to choose and plot the most appropriate distribution in R?

I need to choose the distribution that best fits my data for different datasets. There are similar discussions here and here, but I am still struggling to find proper solution. My first attempt was ...
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Bounding values of a Dirichlet distribution

Consider $k$ random variables $X_1, X_2, \ldots, X_k$ such that $(X_1, X_2, \ldots, X_k)$ follow a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution. For a large enough $k$, I am trying to bound/find ...
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Does a general applicable transitive relation exist between test statistics and p-values?

For a proof I showed that, if a p-value has a uniform distribution between 0 and 1, then transforming these p-values back using the inverse of standard normal results in a random variable with ...
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Let $f_X(x)= \frac{x^2e^{-\frac{x^2}{2}}}{\sqrt{2\pi} } 1_{(-\infty,+\infty)}$, what is this distribution called?

Let $$f_X(x)= \frac{x^2e^{-\frac{x^2}{2}}}{\sqrt{2\pi} } 1_{(-\infty,+\infty)},$$ is the pdf of random variable $X$. What is this distribution called? I only know $f_X(x)\propto x^2e^{-\frac{x^2}{2}} ...
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Example of two Random variables which are independent but from different distributions [duplicate]

I was reading about meaning of iid(independent but with identical distribution). Can there exist some 2 random variables which are independent but from different distributions? Any example?
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Ways to quantify the (dis)similarity between spatial patterns (2D distributions + spatial autocorrelation)

So I have several 10000*10000 grids of the same shape & resolution, and for each grid, grid-points on this grid are discretely valued from 0 to 10 (to be more specific, the value represents the ...
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Dimension changes when taking conditional expectation

I am trying to derive a formula of a paper, and quite stuck at how they did that. Hopefully, someone can help me. So as we all know, for a $d$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}...
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Distribution of sum of squared deviations

So, given $X_1, \ldots, X_n \stackrel{iid}{\sim} N(\mu, \sigma^2)$, I want to find the joint distribution of $\bar{X} = \frac{1}{n}\sum_{i=1}^n x_i$ and $S^2 = \sum_{i=1} (X_i - \bar{X})^2$ (it is ...
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Custom distribution function

I am looking to build a two-parameter distribution function $f\left(x\mid m,v\right)$, potentially with bounded support. For this distribution, the expected value only depends on $m$ and the variance ...
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Non-central correlated normal ratio - distribution of the ratio of two dependent normally distributed variables

I am currently trying to solve a problem in the context of a Bayesian analysis that concerns normal distributions. The situation is as follows. I have an equation that looks like this, where I know ...
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What is Relative Equilibrium Distribution

Some datasets collected are imbalance in nature which violates the assumption of relative equilibrium distribution for most classifier learning algorithms, which will reduce the performance of ...
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Distribution of the ratio of a Normal distribution divided by Lognormal distribution

I want to know the distribution (and the moments) of a variable, $Z = X/Y$, where $X\sim \mathcal{N}(\mu_{x}, \sigma^{2}_{x})$, and , $Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$? Hence, what I want ...
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How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$?

I would like to sample from a distribution on a specific domain $[a, b]$ with mean $c$ where $a < c < b$. Ideally, I would also like the mode (i.e. the "peak") of the distribution to ...
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Parameterized probability distribution for finite, discrete values?

Sorry if I don't have the right terminology for asking this question in a good way ... I'm curios if there is an established distribution function for the following case: I have 20 different options, ...
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What is this metric called? log(var)/log(mean)

I have come across this is a piece of code but I am struggling to find something similar with a statistical explanation. It seems to be used in this way? $$p = \log(\text{var})/\log(\text{mean}).$$ ...
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Identical Random Variables

I am reading the book "Probability - for the enthusiastic beginner" by David Morin. The book makes the following statement about Identical random variables Xi. " The sum X1 + X2 + X3 + ....
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Identify Poisson or Exponential Distribution and determine lambda

I am trying to identify the distribution of my variable, $X$. It measures goals per minute of soccer players. Possible values are $[0,inf]$ and they are non integers. I believe this to be an ...
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Wilcoxon signed-rank test with large dataset and non normal distribution

While the Wilcoxon signed-rank test in general doesn't assume any distribution, most exact implementations are restricted to <50 samples (i.e. scipy). Above that a normal distribution is assumed to ...
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Correct notation for the probability of an event in entropy

I am looking at the formula of entropy on Wikipedia, where $P(X)$ is a probability mass function. \begin{equation} H(X) = -\sum_{i=1}^{n}P(x_i)log_bP(x_i) \end{equation} I got curious why they use ...
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Estimate parameters of a concrete categorical mixture model (information retrieval)

Let $f_{i,d}$ be the frequency of the word $i$ in the document $d$ and $l_d$ be the length of the document $d$. Then $P(X = i \mid D = d) = \frac{f_{i,d}}{l_{d}}$ is the probability of drawing the ...
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How many neglected samples when drawn with replacement? (bagging)

I learned a while ago about an interesting place that $e$ shows up in probability: if there are $n$ items and you sample $n$ times with replacement, you would expect that the fraction of samples that ...
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distribution of maximum random walk distance

Related to this question. Suppose I flip a fair coin $N$ times and keep track of the difference between the total number of heads and tails as I am doing it. At the end of the $N$ coin flips, I have ...
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About transformation of Variables

In a probability chapter of a Python Book, there is the following problem involving a transformation of variables: I don't fully understand where the value 1/z+1 in Y > X(1/z+1) comes from, and ...
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Cumulative distribution function equals almost surely

Let $F_1, F_2$ - two continuous CDF. if $F_1 = F_2\quad F_2$ almost surely (i.e. probability of $x$ where $F_1(x)\neq F_2(x)$ is zero with respect to probability with CDF $F_2$). Then $F_1 = F_2$ (...
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How to compute the covariance matrix of a multivariate bernoulli distribution?

Considering this toy example: Let $x$ be a random variable $x \sim \mathcal{N}(\mu_x, \Sigma_x)$ Where $\mu_x \in \mathbb{R}^2$ is the mean vector and $\Sigma_x \in \mathbb{R}^{2 \times 2}$ is the ...

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