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.

learn more… | top users | synonyms (2)

4
votes
0answers
69 views

Is there a Bayesian approach to density estimation

I am interested to estimate the density of a continuous random variable $X$. One way of doing this that I learnt is the use of Kernel Density Estimation. But now I am interested in a Bayesian ...
4
votes
2answers
60 views

Non-informative prior for regression model

I'm looking at p. 355 of Gelman's Bayesian Data Analysis (3rd ed.), for which there is no errata, and I see this: In the normal regression model, a convenient non-informative prior distribution ...
2
votes
1answer
33 views

Discrete analog of CDF: “cumulative mass function”?

We call the integral of a probability density function (PDF) a cumulative distribution function (CDF). But what's the cumulative sum of a probability mass function (PMF) called? I've never heard the ...
5
votes
4answers
396 views

Calculating PDF given CDF

I know that the PDF is the first derivative of the CDF for a continuous random variable, and the difference for a discrete random variable. However, I would like to know why this is, why are there ...
1
vote
1answer
21 views

Laplace transform and density

It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density? ($L_X(z) = E(exp(-Xz))$)
0
votes
0answers
14 views

Let $f(x)$ be the density of $X$. What is $\lim_{n\rightarrow \infty}\mathrm{E}_X[nf(X+n)]$?

The expectation can be written as $\mathrm{E}_X[nf(X+n)] = \int_{-\infty}^\infty nf(x+n)f(x)\,\mathrm{d}x$. The expectation relies on the speed of tail $f(x+n)$ goes to $0$. I posted a related ...
1
vote
0answers
19 views

PDF for ratio of Dirichlet

I have a random vector $\theta$, which was generated by: X1 ~ Dirichlet($\alpha_1$) X2 ~ Dirichlet($\alpha_2$) $\theta$ = ...
6
votes
0answers
82 views

Sum of normal truncated random variables

Suppose I have $n$ independent normal random variables $$X_1 \sim \mathrm{N}(\mu_1, \sigma_1^2)\\X_2 \sim \mathrm{N}(\mu_2, \sigma_2^2)\\\vdots\\X_n \sim \mathrm{N}(\mu_n, \sigma_n^2)$$ and ...
2
votes
1answer
22 views

Scale parameter MLE scheme known but how to find according distribution PDF?

For known location, we can find the scale parameter of a normal distribution by calculating the sum of squared differences to the location, then dividing by n-1 and taking the square root. This is the ...
3
votes
0answers
34 views

How to calculate CDF of g(X)

Let $X$ a random variable with distribution $F_X(x)$ $$Y=g(X) = \left\{ \begin{array}{lr} X-c & : X > c\\ 0 & : -c < X \le c \\ X+c & : X \le -c \end{array} \right\}$$ ...
2
votes
1answer
49 views

Rectifier function of random variable

Let $X$ be a random variable with distribution $F_X$ and density $f_X$. Define $$g(x) = \left\{ \begin{array}{lr} x & : x \ge 0\\ 0 & : x < 0 \end{array} \right\}$$ and let ...
0
votes
1answer
30 views

Identifying subsets for outlier detection in local outlier factor

I am trying to gain better understanding of the idea of local outliers (as discussed in this pdf) and how the function is implemented. Here are the key passages from the pdf: Local outliers: ...
2
votes
1answer
67 views

Non-central scaled Student's t cumulative density function required (alternatively the pdf)

I need to cite the pdf(density) or cdf(distribution function) of a non-central scaled Student's t distribution. There is an article about the non-central Student's t distribution ...
1
vote
1answer
55 views

Probability function of three Random Variables multiplied, solidifying intuition

I ran across this exercise: Let $T$ be a random variable distributed as a $\text{Bernoulli}(p)$, $U$ be a random variable distributed as a $\text{Bernoulli}(q)$ and $W$ be a random variable ...
1
vote
1answer
103 views

Why likelihood is not always a density function? [duplicate]

I try to self-learn Bayesian machine learning (mostly by studying Bishop and Kevin Murphy's books). While working with formulas I was puzzled by the quote that "Note that the likelihood function is ...
0
votes
0answers
35 views

Moment generating function of a distribution

I want to find the moment generating function (mfg) and mean deviation of this distribution: $$f(x,\epsilon,k,\theta) = k\theta^{(1+1/k+\epsilon/k)}x^{(k+\epsilon)}\exp{(-\theta x^k ...
4
votes
1answer
95 views

Computing inverse probability weights — conditional (multivariate) density estimation?

The general version: I need to estimate $f(A | X)$ where $A$ and $X$ are continuous and multivariate. I'd rather do it nonparametrically because I don't have a good functional form in mind and ...
-1
votes
1answer
26 views

Normal Probability Density Function and confusion over how it arrives at an answer [duplicate]

I am confused at how the normal distribution's PDF capable of calculating a density for a single variable. I understand that the CDF probability of an exact continuous random variable $X$ is 0. ...
1
vote
1answer
36 views

How do I compute the density of this distribution

Ferdosi et al define six artificial datasets to test density estimation methods. Part of the fourth dataset is defined as: $$ Uniform(x,y) = [0,100], Gaussian(z) = [M = 50, var = 5] $$ Where $M$ is ...
1
vote
0answers
13 views

Goodness of fit of an approximation of a PDF

I'd like to evaluate the performance of an algorithm which learns to approximate a discrete PDF for a latent variable given some noisy input. Now, is there a standard test to evaluate the goodness of ...
5
votes
3answers
294 views

Let f(x) be some PDF, and F(x) be its CDF. Shouldn't F(x)=.5 give us the expected value of f(x)?

I was playing around in R and have gotten myself very confused about the relation between probability distributions, their expected values, and their cumulative distribution functions. Say we're ...
4
votes
1answer
97 views

Compute pdf of a k-th order statistic

How to compute the density function of the k-th order statistic of a sample of $X_1, X_2, ..., X_n$ random variables distributed independently but not identically (i.e., $X_i \sim F_i$ with $F_i\neq ...
2
votes
1answer
86 views

Difference between two i.i.d Laplace distributions?

What is the PDF of the difference of two i.i.d Laplace distributed random variables? I know that the difference of two i.i.d Normal variables is still the Normal distribution. Since the properties of ...
2
votes
1answer
47 views

What's an intuitive explanation of the F-distribution's PDF?

I have read on this site an explanation of the t-distribution, and was interested to read a similar one of the F-distribution. The textbooks I have read generally do not cover this topic.
2
votes
2answers
117 views

How to get statistical evidence of similar/different evolution from PDFs

Modified question to better explain the context of my problem: I am studying young stars. When a star is born, it is surrounded by a disk of dust called "protoplanetary disk". Planets form in these ...
3
votes
1answer
41 views

Why do the normal and log-normal density functions differ by a factor?

If a random variable $W$ is Normally distributed, then $\exp(W)$ is Log-Normally distributed. However, the pdfs of these two random variables differ by a factor of $\exp(W)^{-1}$. The Normal pdf ...
0
votes
0answers
15 views

How do I analytically transform Wishart to inverse-Wishart?

How do I transform the probability density function of the Wishart distribution into that of the inverse-Wishart distribution? I'd like to know the whole process.
4
votes
2answers
274 views

In R, what does a probability density function compute?

I understand that the probability density function, pdf, of a continuous random variable is the probability of the variable taking on a given value. I am also thought that for a continuous random ...
0
votes
1answer
38 views

Generating functional-form PDF from Max Likelihood Estimation

For the purpose of this question, please consider me a stats newbie. I'm working on a (very fun!) research project which involves estimating a pdf of "personal values" -- i.e. how much a certain ...
2
votes
1answer
47 views

Hazard and density function in survival analysis with discrete time

I am running a survival analysis with descrete time. For that purpose I use the R package survival with this function ...
2
votes
0answers
77 views

kozachenko-leonenko entropy estimation

I'm trying to implement the entropy estimation based on the closest neighbour from Kozachenko and Leonenko but I'm facing a problem I can't solve. The idea is to work in a new set ...
4
votes
2answers
143 views

How to calculate overlap between empirical probability densities?

I'm looking for a method to calculate the area of overlap between two kernel density estimates in R, as a measure of similarity between two samples. To clarify, in the following example, I would need ...
0
votes
2answers
51 views

Marginal and joint distributions of linear combinations of random vectors

Let $X_1,...,X_4$ be independent $N_p(μ,Σ)$ random vectors. Let $V_1,V_2$ be such that $$V_1=(1/4)X_1-(1/4)X_2+(1/4)X_3-(1/4)X_4 $$ $$V_2=(1/4)X_1+(1/4)X_2-(1/4)X_3-(1/4)X_4 $$ I need to find the ...
0
votes
0answers
31 views

Nadaraya Watson model

I am working through Bishop's book PRML. I have been stuck on question 6.18, for a while and would appreciate some guidance. I feel like I am fundamentally not understanding something.
1
vote
0answers
43 views

Orthogonal series density estimation

I am going through this paper Orthogonal series density estimation. I have a doubt in following Assume that the random variable X is supported on [0, 1], that is, P(X ∈ [0, 1]) = 1, and that the ...
1
vote
0answers
49 views

PDF for net displacement of bivariate normal X and Y

this is my first post on StackExchange so I hope I'm posting in the right place. I'm trying to derive the correct probability density function for the net displacement of random variables X and Y ...
1
vote
0answers
24 views

Modeling empirical probability densities

Say we have a classic regression problem in wich we have a numeric outcome along with some predictors, both numeric and categoric. In a typical prediction problem we use to estimate some parameter ...
1
vote
1answer
83 views

Find posterior distribution

Let $X_{1},..,X_{n}$ be a sample from a poisson$({\lambda})$ distribution. Let the prior be ${\pi}({\lambda})=1/{\sqrt{\lambda}}$. Find the posterior distribution. My work: We have ...
3
votes
1answer
62 views

Proof that the weighted sum of $n$ PDFs is a valid PDF

Let $f_i(y)$ for $i = 1, \ldots, n$ be valid PDF’s, and let $a_i ∈ (0, 1)$ be constants, such that $\sum_{i=1}^n a_i= 1$. Show that the function $f(y) = \sum_{i=1}^n a_i\, f_i(y)$ is a valid PDF. If ...
0
votes
0answers
47 views

Cramer's theorem for a precise normal asymptotic distribution

I am working on a homework problem for my probability class: (Cramer Application) A. Let $X_1, X_2, ... X_n$ be a sample from a distribution with pdf $f(x;p) = q^xp$. Determine the MLE of $p$ ...
0
votes
0answers
23 views

multiplication of 2 PDFs

If I multiply the two PDFs, does the variance of the result PDF becomes narrower than the two PDFs always? In other words, if I multiply likelihood and prior to get the posterior, is the variance of ...
2
votes
1answer
74 views

Radial visualization for Probability Density Functions

I saw the following representation in a paper that I was reading. Can anyone can shed any light on how it was developed? this is the Paper - page 34
1
vote
1answer
64 views

R: result of Levene test correct?

I want to use the Levene test to quantify the homo/heterogeneity of the variances of two samples. The density plot looks like this: But the Levene test for the data [...
0
votes
0answers
16 views

Comparing density plots and scoring the combination

I have a set of density plots that contain the distribution of stock prices. Each graph has 5 density plots as follows that shows the distribution of monthly returns based on their ratings - ...
0
votes
0answers
15 views

Estimation of probability density function

How to estimate probability density function when we have set of feature vectors, but we don't know their statistical distribution?
1
vote
1answer
60 views

Beta function approximation of delta function

I have modified the original question. Does beta distribution function $$f(x,\alpha) = \frac{[x^a(1-x)^b]^\alpha}{B(a\alpha+1,b\alpha+1)}$$ where $B$ is the beta function, approach delta function ...
3
votes
2answers
174 views

PDF of Sum of Two Random Variables [closed]

$X$ and $Y$ are uniformly distributed on the unit disk. Thus, $f_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi}, & \text{if} ~ x^2+y^2 \leq 1,\\ 0, &\text{otherwise.}\end{cases}$ If $Z=X+Y$, find ...
2
votes
1answer
41 views

How to extend PDF of normalized sample to original sample?

To calculate the PDF function using Shannon entropy I have scaled my original sample by simply doing $x'=(x-a)/(b-a)$; where $b=\text{max}(x)$, and $a=\text{min}(x)$ and then I found the ...
2
votes
2answers
89 views

Identifying distribution of a variable

Consider a variable that can take both negative and positive values, and that has the following density plot: I am trying to identify the distribution of this variable. The density plot resembles ...
0
votes
0answers
18 views

What is the proper way to compare two estimated densities using sample data?

Say if have a dataset $X \subset \mathbb R^d$. I have two candidate probabilistic models M1 and M2 (e.g., M1 is a mixture of 2 gaussians and M2 is a mixture of 3 gaussians). I want to know which model ...