Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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Characteristic function of linearly transformed random variables with extracted factor

Given is a sequence of independent random variables $X_1, X_2,\ldots, X_n$ with characteristic function $\varphi_{X_i}(t)$. The characteristic function of $Y = \sum_{i=1}^n a_i X_i$, where the $a_i$ ...
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Prove sum of T(Xi) also belong to exponential family [closed]

Suppose we have an independent and identically distributed sample X = ($X_1$, $X_2$, …, $X_n$) from a distribution that belongs to the exponential family. Say f(x; Θ) = h(x)exp{a(θ)T(x) + b(θ)}. How ...
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Finding the distributions of the max and min of random variables from the geometric distribution

Let $X$ and $Y$ be independent random variables following the same geometric distribution, that is $P(X=k)=P(Y=k) = (1-p)p^k, k=0,1,\ldots,$. Let $U=min\{X,Y\}$, $V=max\{X,Y\}$,and $W=V-U$. How do I ...
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Distribution of a fraction of exponential random variables

Let $X_1, X_2$ be independent exponential random variables with common pdf $f(x)=\lambda\exp(-\lambda x), x>0$. How do I show that $Z=X_1/(X_1 + X_2) \sim U(0,1)$? I know that $F_Z(z) = P(\frac{X_1}...
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Convert a distribution of a varialbe to another one

Assume an experiment where the observable is t and is related to another variable E with the following formula: $$E(t)=A/(t-t_0)^...
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Conditioning on random vector

If I have a conditional probability distribution $\mathbb{P}(X|Y)$, with $X, Y$ being random vectors, do $X$ and $Y$ need to have the same size, i.e., dimensionality?
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From marginal distribution to joint distribution with independence

Consider a random vector $(X,Y,Z)$, Let $f_X, f_Y, f_Z$ be the probability distributions of each component. Question: Does there always exist a distribution $f$ for the whole vector $(X,Y,Z)$ such ...
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what is the difference between mixture of two normal distributions and sum of two independent variables

The following denotes a mixture of a standard normal with a normal with the same mean but 100 times the variance: $0.95 \mathrm{~N}(0,1)+ 0.05 \mathrm{~N}(0,100)$ Let Y = 0.95 X + 0.05 Z with X,Z are ...
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Is the CDF of the Mean always 0.5 for all kind of distributions?

Can we say that the value of the cumulative distribution function at the mean F(X< Mean) is always 0.5 for all kind of distributions that are not symmetric?
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Which kind of distributions can decision trees learn (well)?

Suppose I have a classification task in $\mathbb{R}^d$, given by a distribution $P$ and training data $D$. The decision boundary is $$B = \{x \; : \; P(Y=1|x) = P(Y=0|x)\}$$ Assume I am learning a ...
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Quantiles from the combination of normal distributions with python [closed]

I want to get the support value of mixture of gaussian with respect to 1/4 quantile. How can I get this value when I know the means, vars and weights of gaussians
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How to define 2d-quantiles?

I need to group the values of two distributions into n groups of equal size by using quantiles (remark: I know that quantiles do not always result in exactly same ...
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Determine confidence level in guesstimate: gas stations in Germany

I read in the internet about the following guesstimate during a job interview for a trading position: Give an estimate for the bid-ask on the number of gas stations in Germany (with max spread of 10 %)...
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What are the values of the draws from the uniform distribution used to create x? [closed]

Any help is greatly appreciated. Please let me know if there is any other information that is needed.
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Count variables [closed]

Does anyone know if the variable number of children in a regression is a count variable and if the number of hectares of land is a count variable? Thanks in advance
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Modeling cyclic processes when timing is hard

I have a collection of timestamps for a process that is essentially cyclic: $$ t_{n+1}-t_n = k + E_n $$ where $k$ is an absolute (unknown) constant and $E_n$ is an error term, small compared to $k$. ...
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using Jensen-Shannon Divergence for more than two data sets as a distance metric

I know that Jensen-Shannon Divergence for two distribution is symmetric. Is it possible to use JSD as a distance? In other words, is there a version of JSD that satisfies all the requirements for ...
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Cholesterol levels that follow Normal Distribution and 2 simple questions [closed]

Let cholesterol levels of a population be described by a normal distribution $X \sim N(μ=250, σ = 50), \quad P_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-(x-250)^2}{2 \times 50^{2}}}$ I am asked to compute ...
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Generate Random Numbers with arbitrary distributions [closed]

Let $\Omega\subset\mathbb{R}^n$ and $(\Omega, \mathcal{B}(\Omega), P)$ be a probability space, where $\mathcal{B}(\Omega)$ is the Borel $\sigma$-alebra on $\Omega$. Suppose that $P$ is absolutely ...
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If hitting a target has $P = 0,3$, how many shoots to get at least one hit with a probability of $0.9$?

Cheers, I know that hitting a target has a probability of $0,3$, and I am asked to find the number $n$ of times that I have to shoot at the target to get at least one hit with a probability bigger ...
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If every event is trivial (0 or 1 probability), then every random variable is (a.s.) degenerate/constant. Maybe Lebesgue decomposition?

There are these: 1, 2, 3, but I wanna try different ways. Let $(\Omega, \mathfrak{F}, \mathbb P)$ be such probability space with each (event) $E \in \mathfrak F$ having trivial probability. Consider a ...
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Why are rating distributions 'smoother' when there are more players?

Edit: Please reopen assuming the ff isn't resolved: @Scortchi-ReinstateMonica 1 - so 'other post answers this if iid and if not iid then explain why'? 2 - wht exactly is in the other post that ...
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"Statistical Models can NOT Predict Data Significantly Different from the Data used to Create them"

Anywhere within the field of statistics, has the following statement ever been formalized mathematically? "Statistical Models can NOT Predict Data Significantly Different from the (Joint ...
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Conceptual clarification between various distributions over a circle [duplicate]

Let $D$ be a unit disc $|z| \leq 1.$ Consider the following three ways of choosing a random point in $D$: 1.Chose a point $(x,y)$ where $x=\sqrt{r} \cos t,y=\sqrt{r} \sin t$,where $r =U(0,1),t = U(0,2 ...
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what is the probability of sample variance when true variance and true mean is unknown?

Sample Variance by definition is $s^2 =\frac{1}{n-1} \sum{(x_i-\bar{x})^2}$ When the population distribution is normal and true variance $\sigma^2$ is known, Sample Variance follows the chisq ...
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How to sample from a distribution approximated by a Neural Network?

There are a few models already that approximate distributions with a neural network i.e.: energy models define a density function $f(x)= e^{S(x,w)}/Z$ where $S$ is a neural network and $Z$ is a ...
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Finding a distribution of a countably infinite subset of the unit interval that has the least non-uniformity? [closed]

I want to find an alternate approach to this question. Suppose $f:A\to\mathbb{R}$ where $A$ is a countably infinite subset of the unit interval. I want a distribution of $A$ with the least non-...
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1answer
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Calculating the accuracy of an estimate when the true value is unkown

This is the context for the problem I've run into: A program starts counting down in seconds, choosing a random value with a min of 300 and a max of 600. At the end of the countdown event A occurs, a ...
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On the avaerage modulus of a random polynomial with normal coefficients

I asked this question https://mathoverflow.net/questions/413803/the-average-value-of-the-modulus-fz-of-a-polynomial-with-real-coefficients?noredirect=1#comment1061121_413803 mathoverflow a question ...
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Effective Sample Size for a cycle or mixture of kernels

The Effective Sample Size (ESS) of a univariate Markov Chain can be used to assess it performance. Is there a version of the ESS for when the Markov Kernel is a mixture $\alpha K_1 + (1-\alpha) K_2$ ...
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1answer
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Normal Distribution - Implications of negative values

Consider a task, whereby we need to generate a normally distributed data set of 100 numbers. "tableFreq" is being used to hold the "frequency" data. The sum of the frequencies ...
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Is there any methodology to generate the non negative and non normal random data based on descriptive statistics of probability of lossing the game

I would like to generate non-negative random data (between 0 and 1) and non-normal. Is there any methodology to generate the random data based on the below distribution? Below are the descriptive ...
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1answer
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What is the meaning of "loc" and "scale" for the distributions in scipy.stats? [duplicate]

I need to know the meaning of the variables loc and scale of the distributions in scipy.stats...
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1answer
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Are $U + V$ and $UV$ independent when $U,V$ are independent and standard uniform?

This is related a previous question I posted on the product of two independent variables here. As an alternative method, one could note that if $X,Y \sim U(-1,1)$ and $U,V \sim U(-1,1)$ then $Z = XY = ...
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pdf of the product of two independent uniform random variables $X,Y \sim U(-1,1)$ [duplicate]

Using the product distribution. I have $Z = XY$ with $X,Y \sim U(-1,1)$ and independent. Thus \begin{align} f_Z(z) & = \frac{1}{4} \int_{-1}^1 I_{[-1 < \frac{z}{x} < 1]} \ \frac{dx}{|x|} \\ &...
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Calculating chance of certain score in golf

Im trying to calculate the probability of getting a certain scores on a given golf course when i know the average score the player gets on each hole. I also know the distribution of eagle birdie par ...
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Inference on a selectively revealed sample

I think this question may be related to cryptography, so I may have the wrong stack exchange, but I am not really sure. Suppose there are two people Sam and Pam. Suppose we have a distribution, a set ...
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When is necessay to use "loc" and "scale" for distributions in scipy.stats? [duplicate]

I need to fit some data (about 1000) to a several probability distribution functions (p.d.f.) in order to compare which one fits better. In scipy.stat I can found several p.d.f. and all of them have ...
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Simulate a tournament with N teams

Based on a programming problem I saw recently, I am wondering if it is possible to simulate the following situation more efficiently: There are $N$ teams playing in a tournament. Each team plays ...
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Gradient of Mixture of Normals in log scale [migrated]

I have a mixture of two normal distributions with mixture parameter $$ \pi(x) = \alpha \mathcal{N}(x; \mu_1, \Sigma_1) + (1-\alpha)\mathcal{N}(x; \mu_2, \Sigma_2) $$ What is the gradient of the log ...
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How to show that this integral of the normal distribution is finite?

Numerically, I have noticed that $$\int_{-\infty}^{\infty} \dfrac{\phi(x)^2}{\Phi(x)}dx < \infty$$ where $\phi$ and $\Phi$ are the standard normal pdf and cdf. However, I do not see how to prove it....
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Compare quality of fit, MLE vs method of moments

I have many different datasets that presumably should follow the same distribution type (but with distinct parameters). I've identified one distribution type that seems to describe best the data (...
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1answer
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How to solve this [closed]

There is this problem from past exam papers that I am trying to solve and I can't any ideas? In a bet between two runners the winner will be the one who wins the other 3 times. If the terrain is wet, ...
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What summary statistics are best to recreate a distribution?

I am working on a product where some sensors capture data at 500Hz (500 observations per second) and some sensors capture data at 30Hz. Both are connected to the primary device which sends events back ...
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1answer
156 views

Is there any connection between these two distributions?

I was playing with standard uniform distributions where I noticed a "weird" relation between two combinations and was wondering if there was an underlying reason for it (or if it is just ...
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1answer
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How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?

May I please ask the community's support with the following problem? I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
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Is it true that every uniform random variable is a linear combination of other uniform random variables?

I have the following question to do. Suppose that $G$ is a field and $Y$ is a random variable defined over $G$. Could the following be proved: For every uniform random variable $Y$ defined over $G$, ...
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How to properly apply Benford's Law to apartment service charge data?

I want to apply Benford's Law to a breakdown of billing information we recieve for our apartment building's service charges. In other words, I want to compare the distribution of my service charge ...
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1answer
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Geometric Distribution: Playing a Game with Cost

Suppose I have \$3,000. I play a game, for which I have a probability $p$ of winning. I have to pay \$300 each time I play the game. If I win, then I earn a payoff of \$500. I can play the game as ...
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How can I calculate a confidence metric for selecting the most frequent value in a distribution?

Let's say that I have a dataset of 100 whole numbers, and they all range from -10 to 10. I want to determine the most frequently occurring value, and calculate an associated confidence metric to ...

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