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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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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123 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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24 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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45 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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21 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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15 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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199 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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33 views

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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17 views

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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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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49 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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215 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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0answers
11 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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2answers
25 views

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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34 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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72 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
19 views

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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31 views

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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41 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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22 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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107 views

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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34 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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31 views

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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31 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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88 views

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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32 views

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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1answer
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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33 views

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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32 views

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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11 views

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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30 views

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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274 views

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 ?
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95% region from a bivariate density

I have a bivariate data which can be displayed as a scatterplot of which marginal distribution is Uniform(0,1). Using 'kde2d' function in R, I can also obtain two-dimensional density of these data. ...
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31 views

Consistency of density estimation under marginalization

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be samples from some unknown distribution $p(X,Y)$ and $\hat{p}(X,Y)$, $\hat{p}(Y)$ density estimates of the joint and marginal distributions (i.e., for the estimation ...
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3answers
62 views

Intuition of pdf of a continuous random variable [duplicate]

What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if ...
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1answer
45 views

Linear transformation of a random variable by a tall rectangular matrix

Let's say we have a random vector $\vec{X} \in \mathbb{R}^n$, drawn from a distribution with probability density function $f_\vec{X}(\vec{x})$. If we linearly transform it by a full-rank $n \times n$ ...
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Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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1answer
30 views

marginal conditional distribution from MCMC output [duplicate]

I have a MCMC sampler that targets $$\mathbb{P}(U_1,U_2,...U_n \mid G(U) \leq 0)$$ where $U=(U_1,U_2,...U_n)^T$. I realize now I am more interested in estimating the conditional density $$p_k = p(u_k ...
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40 views

Density Function Estimation

Given a sample of $n$ observations, which are assumed to be $i.i.d.$ and generated from a continuous probability law. Consider the question of estimating the density function $f(x)$. There are two ...
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162 views

Show that $\min(U,1-U)$ and that $\max(U,1-U)$ are uniform

Let $U$ be uniform on $(0,\ 1)$. Show that $\min(U,\ 1-U)$ is uniform on $(0,\ 1/2)$ and that $\max(U,\ 1-U)$ is uniform on $(1/2,\ 1)$. I'm not sure how to approach... the only hint i have is that a ...
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102 views

Finding the Mean and Variance from PDF

A random variable $n$ can be represented by its PDF $$p(n) = \frac{(\theta - 1) y^{\theta-1} n}{ (n^2 + y^2)^{(\theta+1)/2}}.$$ $\theta$ is a positive integer and $y$ is a positive parameter. If ...
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“The total area underneath a probability density function is 1” - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I ...
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1answer
62 views

Get probability distribution function from density function

For a given density function, how does one find its distribution function? For example, I have a density function: $f(x)= \begin{cases} t ^2 / 9 & \text{if } t \in (0,3)\\ 0 ...
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1answer
34 views

Interpretation of cartesian product of the support of marginal distribution

Suppose we have a multivariate data set, $s = (s_1, s_2, ... s_p)$ and each $s_i$ is distributed with a distribution that has finite support (we'll call each $s_i$ a "source"). Let us denote the ...
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19 views

On transformations of random variables, discrete vs continuous [duplicate]

Suppose we have a discrete r.v. $X$, take $Y = g(X) $ where $g$ is one-to-one and onto- If we want to obtain the new pdf for the discrete r.v. we simply notice that $$f_Y(y) = P(Y=y) = f_X(g^{-1}(y)) ...
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198 views

The probability of a random variable being larger than a sequence of random values

Suppose we have a fixed, known, $n$, and each $x_1 \ldots x_n$ is a random number generated uniformly over $[0,1]$. What is the probability that $x_n$ is the largest value in the sequence?
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1answer
37 views

PDF of mixture of random variables that are not necessarily independent

I am trying to derive the expression for the PDF of a weighted mixture of n random variables. Let us taken $n=3$ and define $$X = \alpha_1 S_1 + \alpha_2 S_2 + \alpha_3 S_3$$ $$E[X^2] = 1$$ $s_1$, ...