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Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

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Applying assumptions about marginal and conditional PDFs

We are given $0 < x_2 < x_1 < 1$. What assumptions can you make about $f_1(x_1)$ and $f_{2|1}(x_2|x_1)$? I know that $f(x_1) f_{2|1}(x_2|x_1) = \frac{1}{x_1}$. I know the expression can be ...
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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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Is it possible to find the joint distribution of a random vector if only the distribution of scalar many-to-one transformation is known? [duplicate]

Theoretical Exercise: I'd like to derive the joint distribution $p_{\boldsymbol{X}}$ of a random vector $\boldsymbol{X} \in \mathbb{R}^K$ if only the distribution of a scalar many-to-one ...
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1answer
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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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How to integrate probability density of sum of two indepedent random variables with a finite lower bound on one of them?

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-u^2/2}\:du=1$$ but $$u = \ln(A)-C-k$$ where $\ln(A)$ and $C$ are normally distributed independent random variables, and $k$ is a constant. I am ...
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How many times must I roll a die to confidently assess its fairness?

(Apologies in advance for use of lay language rather than statistical language.) If I want to measure the odds of rolling each side of a specific physical six-sided die to within about +/- 2% with a ...
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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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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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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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Finding probability of a point using bivariate copula density

I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
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What is “data distribution”? Does it Belong to Probability Space?

While reading the paper BEGAN : Boundary Equilibrium Generative Adversarial Network the autor writes as following: "the generator $G(z)$, which maps a sample $z$ from uniform distirubiton to the data ...
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Computational complexity of sampling from discrete and continuous distributions?

What is the computational complexity of sampling from any of these cases? I mean the computational complexity of the most efficient existing algorithm, not a possible algorithm or a lower bound. ...
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Drawing density plot in R [duplicate]

I drew a histogram of my data: ...
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13 views

Generate PDF from CTMC

I have an irreducible continuous-time Markov chain (CTMC) with a finite state space. The CTMC also does not have any one-step transitions from any state to itself. I have the transition rate matrix $Q$...
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26 views

how to get joint pdf of mixed random variables

I would like to know how the joint probability density function $p(b,r,\sigma^2)$ can be calculated for the following graph. Random variable $b$ is a latent binary variable, and random variable $\...
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Hybrid density based clustering using a distance matrix

I want to do clustering in my dataset in SAS for which I want to make following choices in my model: 1) Use a distance matrix which treats categorical variables and continuous variables differently ...
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37 views

Statsmodels' Negative Binomial: after .fit_regularized(), how to turn PMF into PPF to get the discrete values?

I used the package statsmodels to fit a Negative Binomial to my data. This data contains ~1500 samples with 21 covariates. Since I have overdispersion in my data ...
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More general conditional PDFs

The definition of the conditional PDF of X, given Y=y, is well known. How would one define the conditional PDF of X, given X>A (which I believe is called the Hazard Function), in a formal manner? I ...
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pmf for coin toss

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up. Let $x=0$ represent a 'heads' outcome and $x=1$ represent a 'tails' ...
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What is the height of a triangle if base and diagonal of bigger triangle are known? [migrated]

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up. Consider this figure - a rectangle with height 1 and width 2 with ...
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30 views

How do I compute these values from this PDF?

I can not figure out how to find $P(1.9\leq|X|\leq3)$ when $1.9$ is not given as any value of $x$. Would that just be the $P(X=3?)$ I did what I knew how to do, but I am not sure how to proceed from ...
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24 views

How can I find this constant?

My friend asked this question in our class: let X be a random variable which has a cumulative distribution function . Find (a). I think (a) cannot be solved but my other friend thinks (a) = 5/8 ...
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How to estimate probability density function (pdf) from empirical cumulative distribution function (ecdf)?

The context is survival analysis, where I have an empirical survival function in the form of a step function, which is just one minus the ecdf. Is there a standard way to get an estimate of the pdf (...
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46 views

Deriving Posterior Binomial Density from Uniform Prior

I'm trying to derive the posterior density of the probability parameter of a binomial random variable, given one realization of the random variable and a uniform prior density on the probability ...
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2answers
49 views

Finding pdf with more than one random variable

I am stuck with a question doing one of my stats tutorial and question is as follows: Suppose X and Y are two independent exponential random variables with parameter $\theta$, i.e. their joint ...
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A symmetric iid process

Let $X_1, X_2, \ldots$ be an iid process with $X_i$ having a symmetric distribution around $0$. Then can I always write $$X_1 - \alpha X_{t-1}-\alpha^2 X_{t-2}-\cdots \stackrel{iid}{=} X_1 + |\alpha| ...
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47 views

Difference between probability density functions and sampling distributions

I was wondering what is/are the fundamental difference(s) between a probability density function for a mean and sampling distribution of a mean? Can we say that ...
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1answer
39 views

How do you check that a sampler and a density correspond to the same random variate?

General Question If someone handed you a direct sampling algorithm and a density function, and they told you that the two corresponded to the same random variate, how would you check this? ...
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Data distribution without density?

I was reading the Wasserstein GAN paper and in the introduction, the author claims the following. Rather than estimating the density of Pr, which may not exist, we can define a random variable Z ...
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1answer
123 views

What is the distribution of a sum of identically distributed Bernoulli random varibles if each pair has the same correlation?

What is the distribution of a sum of $n$ Bernoulli random variables, each having success probability $p$, where each pair is correlated with correlation coefficient $\rho$? $$Y = \sum_{i=1}^n X_i$$ $$...
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33 views

Why does the PDF use a different variable than x?

In the below image (from Wikipedia but also found in my text book), I noticed that the variable within the integrand is a "u" rather than the "x" which is found in the CDF function. Why is the ...
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What is the expected fraction of observations in the top x% that remains in the top x% after a random shock?

I'm struggling with a probability question, and I was hoping someone here could help me out. Here's the setting. Suppose you draw observations from a probability distribution Z and sort these ...
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100 views

If $X=Y+Z$ with known pdf of $X$, are $Y$ and $Z$ unique?

Say there are random variables such that $X=Y+Z$ with $Y$, $Z$ independent; knowing the pdfs of $Y$ and $Z$, one can (technically) find the pdf of $X$. Taking it from the other side: if one knows the ...
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152 views

When we take draws from a normal distribution what are we drawing? [closed]

As I dig deeper than surface level in probability I'm starting to ask more questions I never thought about before. There are a bunch of intertwined concepts that are quickly becoming confused in my ...
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“Natural” finite measure over continuous probability densities over the interval $[0,a]$ [closed]

I wonder whether there is a "natural" finite measure $\mu$ (such as the Lebesgue-Measure on $\mathbb{R}\cap[0,a]$) over the space of all continous probability density functions on $[0,a]$. EDIT: As ...
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From log-normal parameters, to normal parameters

from the following log-normal fitting function (https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html), I get the parameters [s, loc and scale]. How can I use them to get the μ ...
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Getting weirdly small cdf and pdf values for a set of data of 5 members in R

I am doing a Weibull and normal distribution analysis for a set of my data which are : 336256 620316 958846 1007830 1080401 So to avoid putting the whole code here, I refer you directly to the ...
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Generating random correlation coefficients (Pearson $r$)

I'm trying generate some random correlation coefficients ($CC$) using the Fisher's $z$ transformation. An R implementation is shown below. However, it looks like ...
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1answer
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How to paramaterise a known distribution with mean, standard deviation and fixed upper and lower bounds?

I am looking for something resembling the normal distribution but which is capped at 0 and some size N. The average can be at any point between 0 and N, and there exists a specified standard deviation....
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1answer
58 views

Probability Contour Plot in R

How can I make a contour-plot (of a self-defined pdf) which will contain $25\%$ of the mass within? I was trying to use contour and ...
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2answers
301 views

What is the ratio of a N[0,1] and U[-1/2,1/2]?

I have come across a problem where I can reasonably assume that the numerator is a uniform distribution of the type U[-a,a], i.e., centered on zero, and the denominator is N[0,b]. This seems to be ...
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1answer
26 views

What is the asymptotic distribution of the integrated MSE of the histogram for a discrete random variable?

Let $\{X_i\}_{i=1}^n$ be i.i.d. discrete random variables. Let $f_n(x) = \frac{1}{n}\sum_{i=1}^n \mathbb{1}(X_i=x)$. I am interested in the asymptotic distribution of $$\sum_x (f_n(x)-f(x))^2$$ I've ...
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Joint distribution of multivariate normal

Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$. \begin{align*} P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\ &...
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2answers
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Calculating multivariate integrals between lower and upper bounds

Suppose $\vec{X}=(x_1,x_2,...,x_n)$ follows some continuous multivariate distribution, such that $x_i\in{\rm I\!R}, i=1,...,n$. Suppose also that I have access to the following functions: $\phi(\...
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How to simulate from a gaussian conditional density?

Here's the problem: Suppose that $x$ and $y$ are random vectors which are jointly normally distributed with density $p(x,y)$. I wish to draw samples from the density $p(x|y)$. Denote a draw from $p(...
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The distribution of the initial point of an AR process

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$. Can I say that the distribution of the initial point, $X_1$, ...
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Predicting probability distribution of value in time series of real numbers like Dow Jones?

While we are usually interested in predicting values of time series, it is often also valuable to predict probability distribution of the next value basing on its context - for example for risk ...
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2answers
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Log-normal density function using rlnorm() in R

I tried to draw a log-normal density function by generating random numbers in R. However, the function is not working how I think it should. I draw two similar distribution using two different sample ...
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Strongly rayleigh probability distribution

It is known that the determinantal point processes $DPP$ are special cases of strongly Rayleigh measure $SR$. Could we consider that the permanental point processes $PPP$ are also special cases of ...
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Variable substitution in joint probability density function [duplicate]

Given two continuous random variables $X$ and $Y$ (they can be dependent) and joint probability density function $f_{X,Y}(x,y)$. The question is how to find joint probability density function $f_{X,Z}(...