PDF stands for Probability Density Function. The PDF of a variable gives the relative probability for each value of a continuous variable. Use this tag when asking about probability functions in general, whether PDFs or discrete probability mass functions.

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30 views

PDF/CDF of max-min type random variable

For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as $$F_t(t)=F_{t_i}(x)^n$$ and the CDF of $x=\min(x_1,\cdots,x_N)$ as $$F_x(x)=1-(1-F_{x_i}(x))^n$$ When we have $X=\...
1
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0answers
20 views

Absolute moment of Laplacian distribution

I need to compute the absolute moment of a Laplacian distribution. I just see in wikipedia that If X ~ Laplace(0, b) then |X| ~ Exponential(b ^ −1) where for exponential distribution $$E[X^n]=\...
3
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1answer
14 views

KL divergence vs Absolute Difference between two distributions?

Why should I use KL divergence over just giving the abs difference from two PDFs?
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8 views

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{...
-1
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1answer
37 views

Computing the mean and variance of the ratio of two Laplace variables

I know that Laplacian distribution function is defined as follow $$ f(x)=\frac{b}{2}\exp(-b|x-\mu|) $$ Also, I know that the mean and variance for the ratio between two normal variables like $$c=\...
0
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24 views

Does a probability density function fit here? [closed]

The following is a simplified hypothetical version of my real issue: I'm writing an app that tracks shoe brands in the world using a grid system. The intention is to create a heat map for each brand, ...
0
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0answers
21 views

Conditional CDF and PDF equivalence

Imagine there are two random variables $X$ and $Y$, with CDFs given by $F(x,y)$, and corresponding marginals $F(x)$ and $F(y)$. There exists a PDF, given by $f(x,y)$, with marginals $f(x)$ and $f(y)$. ...
0
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1answer
18 views

Moments from pdf

I got confused reading about moments and their relationship with the pdf. Given a pdf and the values of the parameters, can we calculate the moments of the distribution? More importantly, what is the ...
1
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1answer
34 views

Probability distribution estimate of target continuous variable

I am looking for litterature/reference on algorithms for a regression task that can give the probability distribution estimation of the output variable, or multiple outputs with their respective ...
1
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1answer
26 views

Quantify compatibility between posterior estimates

I performed two distinct, independent experiment $E_1$ and $E_2$ to ideally measure the same quantity $X \in \mathbb{R}$ of interest. For each experiment, I computed the posterior pdf of $X$ via MCMC, ...
1
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1answer
16 views

Distance Metrics for Comparing Theoretical Sampling Distributions

I have a sampling probability density function (denoted $f(x|M,N)$ that becomes hard to calculate for large degrees of freedom (i.e. as $M$ and $N$ get big). If I have another pdf (say $f^{*}(x|M,N)$ ...
1
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1answer
29 views

Is it possible to obtain joint PDF of set of variables given marginal PDF of each variable?

Say $f(X_1)$, $f(X_2)$, $f(X_3)$, $f(X_4)$ are the empirical marginal PDFs of random variables $X_1$, $X_2$ , $X_3$, $X_4$. Also given is correlation between each pair of variables $X_1$, $X_2$ , $X_3$...
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0answers
10 views

Maximization of a special log-likelihood function

I'm clear on how you found the likelihood function by multiply the pdf of all observations and then do the log to help when you derive. But here I don't understand the (2). Is it in a tobit censored ...
7
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2answers
220 views

Best way to evaluate PDF estimation methods

I wish to test some of my ideas that I think are better than anything that I have seen. I could be wrong but I'd like to test my ideas and vanquish my doubts by more certain observations. What I have ...
2
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0answers
15 views

Spinograms vs. conditional densityplots

I have a binary response variable (hail) and multiple continuous predictor variables. My aim is to understand the linear/non-linear relationship of the predictors to the response to be able to justify ...
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0answers
21 views

Joint distribution of density forecasts

I have a panel data set and I have created a model and finally I have obtained some density forecasts. That is, I run my model for the $y_{it}$ and i obtain predictions for $\hat{y_{i,t+1}}$ , $\hat{...
3
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0answers
35 views

Problem involving P.D.F. containing an indicator variable

Let $X_1, X_2, \ldots$ be independently and identically distributed random variables with probability density functions: $$f(x) = \alpha \;x^{-(\alpha+1)} \; I_{(x>1)}, \; \; \alpha > 0.$$ For ...
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7 views

Prediction whose distribution is shifted and is more leptokurtic

I have a model that, based on subject matter knowledge, should give predictions which have about the same density distribution as the training data. The actual predictions have a similar shaped ...
1
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2answers
31 views

Is it better to look at ECDFs than PDFs when exploring empirical sample distributions?

I know that when plotting CDFs, ECDFs, or PDFs by using finite samples, we must be doing some form of interpolation. As far as I guess, empirical cumulative density functions (ECDFs) perform linear ...
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0answers
18 views

PDF of a random variables polynomial

I am trying to study the distribution of the variable $$ F=F_X(X,Y)=f_0+f_1X+f_2X^2+f_3Y+f_4Y^2+f_5XY $$ where $X$ and $Y$ are independent gaussian variables and the $f_i$'s are constants. Is there a ...
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45 views

Find the density of a function of a random variable with known distribution

For example, if $X$ is a normal r.v than the distribution of $Z=X^{2}$ is (F is the CDF): $$F_{Z}(z)=P(Z<z)=P(X^{2}<z)=P(-\sqrt{z}<X<\sqrt{z})=F_{X}(\sqrt{z})-F_{X}(-\sqrt{z}) −P(X=\sqrt{z}...
0
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1answer
30 views

What is the best way to approximate probability from a PDF of a Gaussian distribution? [closed]

In a program I am writing, I have a Gaussian Distribution function that returns the PDF given a specific vector. The issue is, this is obviously not the actual probability. To further complicate ...
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0answers
22 views

Collective confidence of a function - r squared

I have a series of plots, showing time delay against saturation ratio. For each plot I perform a third order polynomial regression to relate time delay to saturation ratio. This generates a function ...
0
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1answer
35 views

What does the height represent in a probability density function? [duplicate]

I know the integral represents the probability, but what does the height represent? It can’t be the number of times that a feature appears, because the function is continuous. I understand that it is ...
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14 views

density plot / kernel

Can anyone explain in more narrative and less technical terms what exactly a density plot is? It seems like it shows the probability of getting a certain value in each histogram bin based on the ...
0
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1answer
28 views

How to decide whether a Kernel Density Estimate is good?

Consider this samples set, feel free to look at it by using your own favourite tools (e.g. R, Python, whatever other stats tools). The problem is that I lack experience in deciding whether any of ...
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1answer
49 views

Finding the expected value of a continuous random varibale when the commulative distribution is given

I have this distribution function of a random variable X: I wish to find E(X). I have used derivatives to get the density function, compared it to 1, and found that f(t) = (4/5)t+(3/5). I then ...
4
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1answer
60 views

Gibbs sampling and conditional distribution

I need to simulate the posterior distribution of intraclass correlation coefficient $\pi(\rho|y)$ where $y$ is the data set and $\rho=\frac{\sigma_a^2}{\sigma_a^2+\sigma_e^2}$ with $\sigma^2_a\sim IG(\...
0
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1answer
40 views

If $W(t)$=$r$+$1$/$s$-$t$($1$-$q$/$s$) how can I calculate the probability density function of $W(t)$?

In this formula t is the time of arrivals (random variable) of vehicles at an intersection and W(t) is estimated delay. How can I develop the probability density function of the vehicle delays given ...
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0answers
20 views

Comparing Two Kernel Density Estimates

I developing a kernel density estimate in Java for a control and test sample population given a certain treatment of the data. I am wondering the best way to test the similarity of the distributions ...
2
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1answer
48 views

Calculating probability of displacement using two CDFs

My knowledge of stats is fairly basic, so you please bear with me! I'm trying to calculate the CDF for the vertical displacement of a (light, small) object floating in a wave tank. I have ...
0
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1answer
23 views

Find parameters of Generalized Inverse Gaussian Distribution

I have a vector of numbers and I am trying to fit the data by Generalized Inverse Gaussian Distribution. My goal is to estimate the parameters $ a,b,p $ which appears in the pdf function. As in the ...
0
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1answer
49 views

Plot the density function of a normal random variable knowing only the characteristic function in R

The characteristic function of a normal random variable with mean $\mu$ and standard deviation $\sigma$ is: $$\begin{alignat*}{1} \hat{\phi}(t) & =e^{i\mu t}e^{-\frac{1}{2}\sigma^{2}t^{2}}\\ &...
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33 views

Multivariate log-normal probabiltiy density function (PDF)

The Multivariate Gaussian pdf is given by $$(2\pi)^{-\frac{K}{2}} \det(\Sigma)^{-\frac{1}{2}} \exp({-\frac{1}{2}}(X-\mu)' \Sigma^{-1} (X-\mu)) $$ The wikipedia for multivariate Gaussians is here ...
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Posterior pointwise uncertainty of multivariate normal-Wishart (variational GMM)

Given a variational mixture of Gaussians (as per, e.g., Chapter 10 of Bishop, 2006), we can compute the posterior predictive pdf: $$ \left\langle p(x|\alpha,\beta,\nu,\mu,V) \right\rangle $$ where $\...
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26 views

Learning distributions in two dimensions knowing that solutions are either horizontal or vertical (within error margin)

I have a dataset, which I plot using scatter plot like this data[['day', 'action']].set_index('day').action.plot(figsize=(10,8), style='.', color='g', alpha=.2) ...
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12 views

PDF from standard deviations plot

I must consider a plot like this that shows the bounds on $x$ (in this case $\sin^2\theta_{12}$), in terms of standard deviations $N\sigma$ from the best fit. How can I extract the PDF of the $x$ ...
3
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2answers
62 views

Which PDF of X leads to a Gumbel distribution of the finite-size average of X?

Consider the statistic "average of $N$ idd random variables $X_i$", $$S_N = \frac{1}{N} \sum_{i=1}^N X_i$$ Consider also that, by a numerical experiment, it is observed that the distribution of $...
1
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1answer
55 views

How to include prior information about target pdf in MCMC

Is there a somewhat principled way to include prior information about a target density $f(x)$ in a sampling (MCMC) algorithm? [This is a much better formulated version of this question, which I am ...
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3answers
3k views

Are CDFs more fundamental than PDFs?

My stat prof basically said, if given one of the following three, you can find the other two: Cumulative distribution function Moment Generating Function Probability Density Function But my ...
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90 views

Sampling from $f(x)$ given approximation $g(x)$

(After some pondering, what I really wanted to ask is how to incorporate prior information about $f$ into a sampling method - see this question.) Suppose you want to draw samples from an (...
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1answer
44 views

How to show $\int_{-\infty}^{\infty}f_{X}\left(x\right)\left(P\left(X<x\right)-P\left(X>x\right)\right)dx =0$

For a continuous random variable X , intuition tells me that $$\int_{-\infty}^{\infty}f_{X}\left(x\right)P\left(X<x\right)dx=\frac{1}{2}$$ and more weakly that $$\int_{-\infty}^{\infty}f_{X}\left(...
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0answers
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Maximizing a non-parametric Probability Density

Assume we have a set of samples and estimate the underlying distribution with a non-parametric density estimator like the Kernel Density Estimator. Lets assume with a gaussian kernel. In my case it ...
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0answers
34 views

Probability of a Point Chosen at Random in the xy-Plane

I am reviewing some of my old basic probability notes and for practice I came across a problem that was provided with a solution, but I think the solution the author provided is incorrect. I am ...
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1answer
42 views

Understanding the CDF of the Exponential from the PDF?

I was trying to get the CDF of the exponential through the pdf. I know that the relationship between the pdf and the cdf is that the pdf is the derivative $ \lambda \exp(-\lambda x) $. But I don't ...
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23 views

Comparison of continuous two-dimensional state space grids across participants

Each participant completed k trials from which I obtained x and y coordinates describing a location on a Cartesian coordinate system (so, each participant has x$_1$, y$_1$, … x$_k$, y$_k$ datapoints)...
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16 views

Cumulative distribution discrepancy

The CDF of a normal variable is $P(X \leq x)$, where $X$ is a random variable. This also written as $\Phi (x)$ so if $\Phi (\cdot)$ is the normal CDF, then $\Phi (0)$ is $P(X<0) = 50 \% $ ...
4
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0answers
42 views

What is the density of a markov chain when its transition probabilities have densities with respect to different measures?

I have a homogenous, discrete time Markov process, $(X_n)_{n\geq 0}$, with state space $\mathbb R_+$. Its transition probabilities have a density, $f(x_n\mid x_{n-1})$, with respect to the measure $\...
3
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1answer
56 views

What's an intuitive explanation for why MAP is variant under parameterization?

I understand why MAP is variant under parameterization mathematically, but I don't really understand it intuitively. To help me out, my professor gave me an example where reparameterizing MAP "...
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22 views

How to rescale data for multivariate t-pdf?

I try to evaluate the pdf of a multivariate t-distribution in Matlab. Unfortunately the function is only defined for a correlation matrix and not for a covariance matrix. I guess i can rescale the ...