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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Characterizing the prominence of a response [on hold]

I’m trying to characterize how prominent a selection is. You poll 10 people for their favorite color and you get the following response: ...
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15 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 \% $ ...
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31 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 ...
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1answer
39 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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10 views

joint pdf of sum and difference [closed]

Let U and V be 2 i.i.d. Uniform(0,1) random variables. Find the joint pdf of U+V and U-V. I'd be grateful if someone gives me the answer to this problem. Any elegant solution will also be welcome.
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14 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 ...
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1answer
352 views

Distribution of a second degree polynomial of a Gaussian random variable

I would like to compute $$P(Y=aX^2+bX+c<0)$$ where $X \sim N(0,\sigma)$. I can do it quite easily using Monte Carlo. However, I've been asked to find the analytical pdf $f_Y(y)$ of $Y$ and then ...
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9 views

Combine Multiple Discrete Probability Density Functions

I'm a bit stuck trying to figure out the combined probability from several discrete PDF's. Lets say I have a bunch of different classes (Truck, Sports Car, Station Wagon, etc) and a bunch of ...
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25 views

Get CDF from PDF

I am trying to evaluate the integral of pdf to get CDF but because of the absolute value, everytime I fail. can someone help me out ? thanks please note this is not HW, it is not practice problem ...
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1answer
20 views

Symmetric PDFs in Metropolis-Hastings

My textbook says that a symmetric PDF satisfies $$f(x|y)=f(y|x).$$ Can anyone explain this? Is it equivalent to $f(x+a)=f(x-a)$?
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1answer
23 views

Standard deviation, quartiles of Arcsine Distribution

I want to compute standard deviation,1st quartile, median and 3rd quartile of Arcsine distribution. I know its PDF is $g(x)=\frac{1}{\pi\sqrt{x(1-x)}},x\in [0,1]$ Its CDF is $\frac2\pi ...
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1answer
25 views

how can I calculate the mutual information between two normal densities using the parameters mu and sigma?

I have two normal densities, $X_1$, $X_2$, for which I know their mean and variance ($\mu_1$, $\mu_2$ and $\sigma_1^2$ and $\sigma_2^2$, respectively). I would like to know the mutual information ...
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1answer
71 views

What is the link (discrepancy?) between these PDF/CDF and p-value distributions?

I have created a mixed distribution model comprising 80% $H_0$ plus 20% $H_1$ to illustrate the link between the expected proportions of true and false positives and negatives in the PDF, CDF and ...
2
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1answer
39 views

Dirac Delta function Notation

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is $\pi(x_{1:n}) = ...
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1answer
54 views

What is the expectation of a Gaussian PDF?

For $X \sim N(x|\mu,\sigma)$ with probability density function $p(x)$, what is $E[p(x)]$?
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18 views

PDF for random variable constrained by other random variables

I'm very novice with prob and stats so maybe my title is incorrect. What I'm trying to do is predict when a set of events will happen in the future. For each event I have a probability model of when ...
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20 views

Understanding Probability Density Functions [duplicate]

I am having trouble understanding probability density functions. What exactly is a probability density function? Does it give you the probability of a value occurring? See my MATLAB code below: ...
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18 views

What tests can I use to compare these two probability distributions

I am trying to compare two one-dimensional distributions. I am using Kullback-Leibler divergence function for this but it requires me to have both the distributions of equal length. I am not sure how ...
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30 views

Bivariate Normal Distribution [duplicate]

Suppose that $(X,Y)$ has the probability density function given below: $f(x,y)=\frac{1}{\sqrt{3}{\pi}} e^{-\frac{2}3(x^2-xy + y^2)}, (x,y)\in \mathbb{R}^2$ a) I want to find the density function of ...
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1answer
913 views

How to calculate the median of a pdf

I'm new to statistics and I'm struggling to solve a question from an assignment. I have a probability density function and I need to calculate its median Here is the function: $$f(x) = 2xe^{-x^2}, ...
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51 views

Joint Density for Renewal Processes

I'm trying to derive the joint density for the time-average age Z and time-average residual life Y for a renewal process, and use that result to determine if Z and Y are independent. If we call ...
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1answer
48 views

Discrete Distribution

In the die-coin experiment, a fair, standard die is rolled and then a fair coin is tossed the number of times showing on the die. Let N denote the die score and Y the number of heads. a)I want to ...
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1answer
47 views

The joint p.d.f of $Y_1$ and $Y_2$ if given p.d.f of $X_1$ and $X_2$

Let $X_1$ and $X_2$ denote a random sample of size 2 from a distribution that is normal$( \mu, \sigma^2)$. Let $Y_1=X_1+X_2$ and $Y_2=X_1-X_2$. Find joint p.d.f of $Y_1$ and $Y_2$ and show that these ...
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17 views

Difference between gaussian and lognormal

I have to study tolerance intervals for a distribution of a random variable Z that is given by the difference of a normal X minus a (independent) lognormal Y. To begin with I tried to get an ...
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1answer
84 views

Multi-variate uniform distribution

Suppose that $(X,Y,Z)$ is uniformly distributed on $\{ (x,y,z) : 0 \leq x \leq y \leq z \leq 1 \}$ a. I want to find out joint density function of $(X,Y,Z)$. b. I want to find out probability ...
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126 views

Probability density of sum of two beta random variables

Suppose that $X$ and $Y$ are independent and have beta distributions. $X$ has probability density function $g(x)=6x(1-x)$ for $ 0\leq x \leq 1$ and $Y$ has the probability density function ...
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10 views

Optimizing a linear combination of time-series to have a certain distribution

Suppose I have $N$ different time-series. I would like to take a linear combination of those time-series, so that the probability distribution function of the linear combination resembles some given ...
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13 views

Is it true that for $t\in[0,1/2]$ and $x\in[t,1-t]$ we have $f(x)\geq f(t)$ if $X$ is single peaked left-skewed with support $[0,1]$

From the typical plots of single peaked and left skewed distributions, I am guessing the statement is true. But is it possible to prove it formally or give a counter example?
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1answer
38 views

Probability distribution of a sine wave

This question on Cross Validated provides an excellent illustration for what I am going to ask. Could you please explain to me why the probability density function of a sine wave looks like it does, ...
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29 views

the functions of random variables

I'm studying for a course and a question came up asking to find multiple functions h such that $Y=h(X)$, where $X \sim$ uniform$[0,1]$ and $Y \sim$ uniform$[-2,8]$. I understand that $f(x) = 1$ when ...
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2answers
74 views

Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables

Are there any advanced results established regarding the behavior of the Covariance of two random variables other than the bounds on the correlation and independence when it is zero etc. which are ...
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35 views

Probability Density Function of a linear combination of 2 dependent random variables, when joint density is known

Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$. What is the PDF of the weighted sum of these two variables, $Z = aX + bY$? Thanks in advance for any ...
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37 views

Joint Density and Covariance between Two Random Variables with the same Mean and Variance

This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this. Q1) Are there any general results / relationships to get the ...
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79 views

Generating random numbers from the skew-t distribution, problem with density plots

in another question I was trying to replicate density plots using random numbers coming from the skew-t distribution of Hansen (1994). Now I need to obtain a series of random numbers coming from this ...
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64 views

Using projected points from Linear Discriminant Analysis to generate probability density function

I am using Linear Discriminant Analysis technique to get the best possible separation between two distributions. I am using R for my programming. For LDA, I find the between-class scatter matrix(B) ...
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6 views

Distibution of the pdf as a random variable when evaluated at samples generated from itself

Let $x_i \sim p$ for some probability density function $p$ with respect to Lebesgue measure on $\mathbb{R}$. Then each of $p(x_i)$ is a random variable taking values in some interval $[0,c]$ from some ...
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1answer
92 views

How to compute the CDF of this random variable?

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from ...
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1answer
31 views

Is an improper prior/posterior equivalent to an undefined PDF?

A "proper" prior or posterior distribution is defined as a distribution for which the PDF integrates to 1 (or in practice, if we're working with a known distribution, one for which the PDF without ...
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Prove f(x) is a probability density function (pdf)

So in order to prove that $f(x)=1/(x+1)^2$ $for$ $x>=0$ and $0$ $otherwise$ is a probability density function, it needs to satisfy the "properties" of a PDF which are: 1) $0<= f(x)$ 2) $\int ...
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1answer
56 views

Difference between joint density and density function of sum of two independent uniform random variables

I am not able to understand the difference between the joint density function and density function for a random variable $Z = X_1 + X_2$, where $X_1, X_2$ are uniform rvs in $[0,1]$. I think joint ...
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1answer
43 views

How can I approximate a pdf knowing the estimated CDF in R?

I have an estimate of a CDF in R (nonparametric) and I need to compare this distribution to another one by Kullback-Leibler. In order to do so, I need to find the pdf of this random variable. What is ...
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16 views

Separable Kernel Density Estimate

Thank you for reading my question. Before I begin, I am no mathematician and so any help/pointers are very welcome. I have been reading the following paper: ...
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11 views

PDF of hourly data

I have data for every hour of the past year for an insurance company making cold calls to sell. Each record indicates the hour and a Y or N if the salesperson was able to make the sell. If I wanted to ...
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1answer
30 views

Probabilistic Density

Variable $X$ has the following probability density: $f(x) =$\begin{cases} 0, & \text{x ∉ [0,2]} \\ kx(2-x), & \text{x ∈ [0,2]} \end{cases} How can I find the parameter $k$ so $f(x)$ is a ...
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1answer
35 views

What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables

$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs. ...
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2answers
45 views

Densities and Cumulative Distribution Functions

Has anyone seen this notation before? What does it mean? $\int_{0}^{\infty} f(x) G(x) dx$ $f(x)$ is a density and $G(x)$ is a cumulative distribution function
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how to measure similarity of two lists of continuous data with different length

I have two lists of continuous data with different length. a) How should I measure (dis)similarity of these two lists? or, as these lists can be formed into histogram, how can I quantify ...
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19 views

Copula density bounds

I was wondering if someone can help me with a problem I encountered in my work. I need a bivariate copula density that meets two constraints at the bounds, and I have difficulties in finding one that ...
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48 views

Generating random variables from a density function expression with R? [duplicate]

I am using R langage, and for my algorithm I should generate random variables knowing the expression of the density function. For exemple, the density function is : ...
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1answer
21 views

Constructing a random Experiment

Can someone help me as how to construct a random experiment which has the following density:$$\frac{1}{2\sqrt{2\pi}}\left(e^{\frac{-x^2}{2}}+e^{\frac{-(x-10)^2}{2}}\right)$$ Update: According to what ...