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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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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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1answer
93 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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1answer
54 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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How to export mulitple PDFs from R [migrated]

I'm fairly new to R and am running into some issues exporting multiple graphs from R into a PDF file. The graphs I've created are: ...
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14 views

How to obtain an approximation to a PDF that cannot be obtained analytically?

Let $W$ be an RV with a Weibull $\mathcal{W}(a,b)$ distribution. Then$^*$ the PDF of $X=\log(1+W)$ is $$ f_1(x) = 10^x \left( 10^x-1 \right)^{a-1}ab^{-a}\ln(10)\exp\left[ -\left(\frac{10^x-1}{b}\...
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62 views

“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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1answer
29 views

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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2answers
155 views

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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0answers
25 views

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
9 views

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
44 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
287 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
24 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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25 views

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
38 views

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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1answer
66 views

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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1answer
52 views

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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0answers
28 views

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 ...
2
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2answers
75 views

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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0answers
28 views

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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0answers
17 views

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}(...
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0answers
20 views

Compute quantiles numerically

I was wondering if there is a way to compute quantiles numerically for distributions were the integral of the PDF is really complicated Any ideas?
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1answer
30 views

Splitting into equal probability intervals

I have some distributions (some of them gaussian and some of them not) for which I know the PDF. I would like to split them into intervals of equal probability. I would like to use the PDF of the ...
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1answer
46 views

Probability of sample point given a Linear Regression

This question may be ill-posed, but hopefully you all can help talk me through it. Given a probability density function $f(\cdot)$ parameterized by one or more parameters $\theta$, we can compute the ...
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0answers
22 views

When is the convolution of symmetric bimodal densities unimodal?

Let $X$ and $Y$ be real valued random variables with densities $f_X$ and $f_Y$. It is well known that if $f_X$ and $f_Y$ are symmetric about zero and unimodal then their convolution $f_X \ast f_Y$ is ...
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1answer
36 views

finding quantiles of a kernel density estimation

I used R to find kernel density estimates of my dataset (for experiment I used 1000 samples generated from a known distribution in this step). I used code density()...
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0answers
12 views

PDFs of two random variables that have an affine relation

Suppose $x$, $y$ are random variables, with pdfs $f_X$, $g_Y$, respectively. If $y = x + k$, for some finite constant $k$, is it the case that $g_Y (y) = f_X( y - k)$? I feel like this is probably a ...
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2answers
56 views
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1answer
25 views

Mean nonequality test if data are intervals

If $X, Y$ are two sets of observations of two random scalar (univariate) variables, one can determine if the expected values of the two variables are unequal, with appropriate tests. My question is: ...
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2answers
443 views

Sample multivariate PDF from KDE with different norm [closed]

I am using KDE with a modified metric for the distance. The PDF is as expected (see below: color is the probability and the dot is the point used to fit the KDE). But due to the new metric, I cannot ...
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1answer
104 views

How to simulate a random variable with this density?

I need to generate random variables $R$ with this distribution function: $$F_R(y)=\frac{1}{\mu(\lambda,\tau)}\int_{0}^{y}e^{-\lambda x^{\tau}}dx$$ where $0\lt\tau\lt 1,$ $\lambda\gt 0$ (it is like ...
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0answers
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How to estimate pdf for sealed bid auction, in which some auctions are unwinnable?

My problem breaks down into two parts: I'd like to use bid data to estimate the probability of winning an auction, based on the bid amount. Assuming only fair auctions, would it be sufficient to get ...
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2answers
37 views

General way to calculate or think about non-linear but monotonic (?) transforms of random variables

I am doing a lot of work with lognormal RVs. I am trying to get my head around the formal mathematics of the non-linear transform of a random variable, particularly where there isn't any '...
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1answer
95 views

Find the PDF from quantiles

I have been presented a problem of this kind: suppose I know the values of k quantiles for a continuous random variable $X$ $$X_{1\%} = x_1, X_{5\%} = x_2, \dots , X_{99\%} = x_{k}$$ so that $$ ...
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1answer
14 views

Different density-functions in different books for Exp(a), why?

This is more of a theoretical question, that I hope someone is willing to explain to me. I have noticed that the density function for the exponential distribution looks different in two of my books. ...
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18 views

High density of predicted probability distribution

I want to predict the likelihood some event happens. The event rate occurs 0.5% of the time globally. I trained on 100k instances and tested on 20k. I built two binary classification models to ...
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0answers
9 views

Estimating conditional pdf using kernel estimator

In a paper I'm reading they consider the logarithmic return $Y$ and the number of trades $T$ over a time periof of fixed size. They then want to show that $Y$ conditioned on $T$ is approximately ...
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1answer
72 views

Must we require the existence of probability density in terms of statistical application?

It is obvious that probability density does not necessarily exist for a given probability distribution. However, many statistical applications assume the existence of density and even use closed form ...
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1answer
36 views

Anyone seen this parametrization of Weibull?

My lecturer uses a parametrization of Weibull that I can't find any where else so I'm wondering are they mistaken. Can anyone confirm if this is legitimate pdf of a Weibull? $$\lambda\theta y^{\...
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0answers
22 views

Getting shape parameters from a beta probability density

Is there a way to calculate the shape parameters $a$ and $b$ of a beta distribution having only its probability density function? This small example might clear a bit what I want (I work on Python): <...
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0answers
17 views

Creating a joint density from two asymmetric uniform marginals

Two RV \begin{split} X_1 & \overset{{i.i.d.}}{\sim}\mathcal{U}[0,1] \\ X_2 &\overset{{i.i.d.}}{\sim}\mathcal{U}[0.3,2.08] \end{split} I need to create a joint pdf from these two. Is copula ...
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2answers
125 views

Uniform PDF of the difference of two r.v

Is it possible to have the PDF of the difference of two iid r.v.'s look like a rectangle (instead of, say, the triangle we get if the r.v.'s are taken from the uniform distribution). i.e. is it ...
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2answers
55 views

Finding a pdf of an exponential distribution [closed]

Random variable $X$ is distributed exponentially with mean 1. Find the pdf of $Y=(X-1)^2$ I'm not quite sure what this question is asking. Could anyone help me out?
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1answer
28 views

Why is the marginal continuous pdf of X+Y in this form?

I read in a book that given a joint continuous pdf $g(x+y)$, for $y$ is: $$ \int_{0}^{\infty}g(x+y)dx = \int_{y}^{\infty}g(x)dx $$ I got stuck here, how did $\int_{y}^{\infty}g(x)dx $ come out?
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1answer
66 views

Finding the discrete distribution

I want to find the discrete distribution of X, where expected value of x, E(X)=3 and the Variance of X is 15. X=1,2,3,4,5,and 6. What is the easiest way to find the distribution of X. I really ...
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1answer
111 views

Finite state machine with gamma distributed waiting times

I have a state machine with positive and negative inputs. The time between positive inputs follows a gamma distribution ($X_+ \sim \Gamma(k_+, \theta_+)$), and the time between negative inputs follows ...
2
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2answers
64 views

PDF for a transformed variable

Let $Y$ have the probability density $f_Y(x)$ and let $X$ have the PDF $f_X(x)$. $X$ and $Y$ are continuous and independent from each other. If $f_Y$ and $f_X$ are known and $Z=g(X,Y)$ where $g$ is ...
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20 views

Let $\xi$ random vector and $\zeta^{x}:=x^{T}\xi$ random variable. Is it correct to say that $\int x^{T}y f_{\xi}(y)dy=\int tf_{\zeta^{x}}(t)dt$?

Let $\xi\in\mathbb{R}^{m}$ be a random vector with joint density function $f_{\xi}:\mathbb{R}^{m}\rightarrow \mathbb{R}$. Let $x\in \mathbb{R}^{x}$ be a nonzero vector. We consider the random ...
2
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2answers
56 views

chain rule in differentiating a CDF

This may be a very easy question but when a CDF is differentiated, it becomes a pdf: $${\partial \over \partial z} F_Z(z) = f_Z(z)$$ But let's say I have a CDF of $F_Z(\sqrt{z})$. Is the pdf this: $${\...
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2answers
44 views

Uniform Density Function

As we know the uniform probability density function is f(x)=1/(b-a) if i find the density function and area of this uniform distribution between (0, 1/2) then it would be f(x)=1/(1/2-0) f(x)=2 ...