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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Imposing a model on a pdf

(This question is an attempt to zoom in on the key issue in this question using as little information as possible.) Lets say I want to derive the likelihood function of $\beta$ given $x$ and $y$ for ...
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Derivation of likelihood function for latent variable model made explicit

I am trying to make the steps deriving the likelihood function for the following latent variable model as explicit as possible: $$Y^0=X\beta + u$$ where $$u \sim NID(0,\sigma^2).$$ The observed data ...
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Estimate density before or after separating into pre-defined bins?

I want to estimate the density of positive real item price data so that I can predict expected revenue per transaction given a known service fee schedule. Suppose also that the fee schedule is defined ...
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47 views

Probability Density Functions of a Worker and his Waking up routine

The Problem: A worker wakes at 6 am and lies in bed for up to 2 hours. Upon rising it takes him an hour to shower and prepare which is preceded by him doing whatever he pleases. He never leaves for ...
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How to extrapolate future probability density functions if you have a time series of them as input?

I'm sorry for lack of technical vocabulary, I'm not a mathematician but an undergraduate student in business informatics. This is my current situation: I am given an observations vector ...
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1answer
40 views

Understanding “the kernel has zero mean”

I am trying to understand kernel density estimation and found the graphic below illustrating different kernel functions on Wikipedia. I have no trouble reconciling it with the two statements "the ...
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10 views

Multivariate extension of Welch-Satterthwaite approximation

The Welch-Satterthwaite approximation can be used to approximate the distribution of sums of gamma random variables. See section 4.1 here, for example. Can we use a corresponding approximation for ...
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11 views

Additive best-fit parametric kernel estimation?

I'm thinking of a process similar to kernel density estimation, but using fitted kernels. The process would be something like this: Create a histogram $H_1(x)$ with infinitesimal bins (such that the ...
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14 views

How does the stats.gaussian_kde method calcute the pdf?

I am using the scipy.stats.gaussian_kde method from scipy to generate random samples from the data. It works fine! What I have ...
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16 views

Interpreting Poisson regression: Deriving results from the CDF

I am trying to interpret a Poisson regression without being very interested in the mean. As a complication, I also have an exposure variable. Let $y_i$ be a count variable, and $p$ the offset, for ...
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27 views

Estimate density dependent on another variable

I have an (unknown) random function $y=f(x)$, i.e. for each value of $x \in [0,1]$ it is a random variable with some distribution. Also I can sample this function, and got values of $y$ for many ...
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55 views

How is $p_{i}$ for a set of continuous data points related to the probability function $f(x)$?

I have the following set of continuous measurements: 155.08 178 264.81 238 378 140.38 130.5 140.69 155.5 To average this data, I sum the values and divide by the ...
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1answer
75 views

Overlap between two normal pdfs [duplicate]

I have two normally distributed random variables (estimated from two different sets of samples), and I'd like to know how "similar" those variables are (in order to compare the sets). I had the idea ...
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1answer
41 views

How to interpret height of density plot

How should I interpret the height of density plots: For example in the above plot, peak is at about 0.07 at x=18. Can I infer that about 7% of values are around 18? Can I be more specific than ...
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9 views

How can I write a skew-normal distribution function given these 3 points?

Suppose I have a set of normally distributed data with mean µ, such that 34% of the data lie between µ–σ and µ, and 34% more of the data lie between µ and µ+σ. Then we know I can write a distribution ...
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1answer
74 views

How to find the normalizing constant for a distribution of unbounded support?

The probability density of a random variable is $$f(x) = ax^2 e^{-kx} ;k\gt0,0\le x\le \infty$$ What is the value of $a$? I understand that first we'll have to take the integral of the function ...
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137 views

Deriving Density Function (pdf) from Distribution Function (cdf)

A random variable $V$ has the distribution function: $$ F(v) = \begin{cases} 0, & \text{for $v<0$ } \\ 1-(1-v)^A, & \text{for $0\le v\le1$ } \\ 1,& \text{for $v>1$ } \\ \end{cases} ...
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25 views

Why small values produce undulating densities when ploting logarithm of a loguniform prior (in R)?

I am using a program that draws random values in a log-uniform distribution let say between 1 and 100. When I plot the density of the produced values with R it looks like a log-uniform distribution ...
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51 views

What is the name of this distribution family?

I am trying to identify this probability density function so I can read up on it to find confidence intervals for $\theta$: $$f(x;\theta,v)=\frac{\theta v^\theta}{x^{\theta+1}}I_{[v,\infty)]}(x);\ ...
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24 views

Is there any general formulation procedure of probability density functions?

There are so many probability density functions for continuous variables around the world. Unlike the probability mass functions of discrete variables, these PDFs do not directly give you the ...
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18 views

Distribution with fixed mean and closest to a given distribution

I was wondering if this problem has been tackled in some way in the probability/functional analysis literature: Given a pdf $f$ such that the expectation is zero and $\mu\in\mathbb R$, find the ...
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14 views

Find the mean of lognormal rv's with available variance and the sum of rv's

I have the sum of a bunch of random variables $S$, v = [1 1 2 2 3 3 4 ...]; S = sum(v); I know that vector $v$ is lognormally distributed, BUT I DON'T KNOW IT. ...
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204 views

PDF of dependent variables

In my recent question an answer was given, and I am able to compute it myself. Still, I'd like to understand where does that answer come from. Hence, what's the approach to handle dependent variables ...
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Compute a PDF in Mathematica/mathStatica [closed]

Let $X,Y$ be iid uniform in $[0,1]$ RVs, and $U$ has a PDF $f_U(u)=\frac{1}{4}\ln\left(\frac{4}{u}\right)$, $u\in(0,4]$. Mathematica itself is able to compute the PDF of $X+Y+\sqrt{(X-Y)^2+U}$ (see my ...
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Multivariate distribution for products of random variables

Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
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What is the name of the density estimation method where all possible pairs are used to create a Normal mixture distribution?

I just thought of a neat (not necessarily good) way of creating one dimensional density estimates and my question is: Does this density estimation method have a name? If not, is it a special case of ...
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101 views

PDF of a sum of dependent variables

This is a direct continuation of my recent question. The thing that I actually want to get is the distribution of $a+d+\sqrt{(a-d)^2+4bc}$, where $a,b,c,d$ are uniform in $[0,1]$. Now, the ...
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50 views

Moments and density tails

Assume that the first $n$ moments $m_1,\dots\,m_n$ of a random variable $X\in\mathbb{R}$ are known, but not its probability density function $p(x)$. Does there exist a methodology to characterize ...
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225 views

What's the distribution of $(a-d)^2+4bc$, where $a,b,c,d$ are uniform distributions?

I have four independent uniformly distributed variables $a,b,c,d$, each in $[0,1]$. I want to calculate the distribution of $(a-d)^2+4bc$. I computed the distribution of $u_2=4bc$ to be ...
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12 views

Student t distribution: standardization [duplicate]

I got a question concerning the standardization of a student t distribution. I see that the "plain vanilla" t distribution has density $f(x|\nu)=\frac{\Gamma(\frac{\nu+1}{2}) ...
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Piecewise-constant density estimation

I came across the term "piecewise-constant density estimation" in a paper and haven't been able to find a definition for it online or in my textbook resources. No example was given in the paper ...
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45 views

Getting the probability density kernel estimator with R

I am working on a density estimation project and I need to get an estimation of the density as well as an equation for the density estimator (and not the estimate). I am working with kernel ...
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73 views

Probability from normal distribution: < vs <=

I want to calculate the probabilities $P\{X < 0.5\}$ and $P\{X \leq 0.5\}$. $X$ is standard normally distributed. From what I have learned density function $\text{df}(x)$ I can get $P(X = x)$ ...
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1answer
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Probability of event depending on a variable with a given distribution

Let $A$ be an event that happens with probability $1-\alpha$, where $\alpha$ has the density function $f_\alpha(x)=3(1-x)^2$ for $0\leq x \leq 1$. Thinking analogous with the discrete case*, I came ...
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Observation Likelihood in hidden Markov models

As far as I understand, in discrete HMM, the observation symbol probability distribution $b_{i}(O_{t})$ is always a probability less than 1, e.g. $\frac{1}{6}$ for each side when rolling a dice. But ...
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47 views

Marginal Probability Density Function of Joint Distribution

I have this question regarding marginal probability density function of joint distribution. Following is the equation I have. $$f(x,y) = \begin{cases} \frac{3}{2} y^2 & 0 \le x \le 2 \text{ and } ...
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23 views

Values of PDF in Bayes Classifier

I'm new here and also a beginner in statistics. I'm implementing a Bayes classifier for two classes but get confused with the value of likelihood (pdf). $$P(c|o) = p(o|c)\cdot P(c)/p(o);$$ Here ...
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In the definition of probability density function, does it matter if the interval is open or closed?

I can find two definitions of Probability Density Functions in the sources I have checked: $$P(a < X < b) = \int^b_a f(x)dx$$ Ref: Hogg & Tanis, Probability and Statistical Inference and ...
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35 views

Likelihood of a Poisson-described event to occur in the next second

Consider a recurring event for which the time periods between consecutive events is exponentially distributed. For argument's sake, I'm waiting for a taxi on a busy street. How might one calculate the ...
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Choosing among PDFs

This is a pretty broad question. I just learned that two random variables can have the same moments but different PDFs. Take $\mathbb{E}[X_i] = \mu$ and $\mathbb{Var}[X_i] = \sigma^2$. Since there are ...
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37 views

conditional probability, change of variable and Jacobean

I have a question, and I am guessing that the question arises due to my lack of good understanding in the change of variable technique. I would like to evaluate $f_X(x)$. When $f_Y(y)$ exists, I can ...
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How to measure similarity of bivariate probability distributions?

I have three different distributions of 2D data: or Now I like to know whether distribution two is more similar to distribution one (2 to 1) than distribution three is to distribution one (3 to ...
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Accounting for measurement bias using a histogram or violin plot and numerical data in R

The Problem I have a gardener whose job it is to measure trees in meters using a measuring stick. She measures the heights of 1,000 trees. I plot these measurements on a violin plot in R. I am ...
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51 views

How to show that the t-distribution density function is a pdf?

We know that the pdf of the t-distribution is : $$f(t|p)=\frac{\Gamma(\frac{p+1}{2})}{p^{\frac{1}{2}}\Gamma(\frac{1}{2})\Gamma(\frac{p}{2})}\cdot\frac{1}{(1+\frac{t^2}{p})^{\frac{p+1}{2}}} \;\;\;\; ...
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When to use CRPS (Continuous Rank Probability Score)? What are the alternatives? What are the advantages and disadvantages?

Please correct me if I'm wrong, crps is new for me. I want to understand it better. We have to minimize crps, which is based on the cumulative distribution function of the data. While the information ...
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Can I use glm with Poisson family if counts data are treated as density?

Imagine you have data of birds counted in an area - let's say, you count 18 birds in a surveyed area of 1,3 km^2. Imagine you relate this counts to 1km^2, so that you have 13.9 parrots per km^2. ...
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How to compute the pdf for logit/probit models?

According to the probit/logit models, the change in probability due to a change in an explicative variable x is given by the following equation: P(Y = 1 |X) = ...
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About the burr distribution 3 parameters

In my papers the probability density for a burr distribution is given as $f(x) = \dfrac{\gamma \tau \alpha^{\gamma}x^{\tau - 1} }{(\alpha + x^{\tau})^{\gamma + 1}}$ however i have encountered this ...
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Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
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Why does a Cumulative Distribution Function (CDF) uniquely define a distribution?

I have always been told a CDF is unique however a PDF/PMF is not unique, why is that ? Can you give an example where a PDF/PMF is not unique ?