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Questions tagged [pdf]

Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

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Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
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Pdf of a deterministic function on [0,1]

I am having a hard time trying to formalize the pdf of a function of a random variable uniformly distributed on $[0,1]$. Formally, I have $Y^n$ is iid distributed according to $\mathcal{U}([0,1])$ , ...
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Distribution of maximum of normally distributed random variables

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far: $$ \begin{align*} F_Y(y) &= \mathbb{P}(\max(...
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The pdf of a standard uniform random variable divided by constant

For a random variable $\frac{U}{a}$ where $U$ is a standard uniform random variable, I'm trying to determine the pdf. I'm not so sure what I'm getting is correct as I'm getting some funny results ...
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34 views

Is there a name for the distribution whose PDF is -ln(x) on its support [0, 1)?

If so, what is its name? If not, how/where can information about it be found?
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How this Equation is solved? How dBi is changed into rdr?

$Y_i = \frac{|h_{B_i}|^2}{1+d_{B_i}^\alpha}$ $d=distance, h_Bi=gain$ $f_{W_{B_i}}(\omega_{B_i}) = \frac{\lambda_{\Phi_B}}{\mu_{R_{D_B}}}=\frac{1}{\pi R_{D_B}^2} $ \begin{align} (CDF) of Y_i .... ...
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Exponential Family Representation: Dumb question on scale parameter and whether it went to Hawaii

So going over the Hastie Tibshirani paper on GAM - it points to equation 11 as the exponential family density - but with two parameters - theta (natural parameter) and phi (scale). https://...
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45 views

Transform X to get Y such that Y has a Uniform(0,1) distribution

A random variable $X$ has the PDF $f_X(x) = \frac{x - 1}{2}, \ 1 < x < 3$ Find a monotone function $u(x)$ such that the variable $Y = u(X)$ has the distribution $Uniform(0,1)$.
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Measure of dispersion around the mode

I usually associate the standard deviation with the mean and the IQR with the median. Is there a measure of dispersion typically associated with the mode?
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61 views

CDF and PDF of radius of a unit disk

Let X and Y be uniformly distributed on a unit disk such that $x^2 + y^2 \leq 1$ Let $R = \sqrt{X^2 + Y^2}$. What are the CDF and PDF of $R$? I know that the area of the unit disk is $A = \pi r^...
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Point process - intensity function vs probability density function

Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have $$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$ where $\Lambda({ S})$ is ...
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How to make recognition of the important document's attributes

We have a set of PDFs with the different types of documents from the various companies. The goal: to predict which of them contain some important attributes (for example, document number, customer ...
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Issue when deriving an expression for normal distribution population parameter PDF given measured sample parameters

So I came across an issue when I derive the equation of the joint PDF for the population mean and variance based on measured sample mean $\bar{x}$ and sample variance $s^2$ and I am not quite sure ...
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1answer
38 views

Histogram and probability mass function

I have a dataset of a discrete random variable. My question is: Is the normed histogram(I divide the frequencies by the total number of samples) and the PMF is the same quantity? It seems they are. Is ...
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1answer
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Visualize Covariance when only probability mass and marginal functions are given

I am trying to intuitively understand Covariance like here. So if a random sample set given, I could draw rectangles with them, one of the cornes being fixated on mean $(\overline{x},\overline{y})$. ...
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Why does the Y axis change in population density plot when changing from raw scores to z-scores (assuming normal distribution) [duplicate]

Why do values on the Y-axis change in a probability density plot when changing from raw values to z-scores? The mean z-score aligns with 0.40 on the Y-axis, while the Y-value for the mean with ...
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124 views

Conditional Expectation of pdf

Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following: $$f(x, y)=\begin{cases} 1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
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Order statistics for log series distribution

I am trying to obtain the probability mass function for various order statistics of a log series distribution for a given n. To do so, I tried modifying the code given in this question: Simulating ...
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1answer
190 views

Median of Rayleigh Distribution

I am not sure how to solve the following problem: The probability density function of the Rayleigh distribution is, $\ f(x;α) = \frac{x}{α^2} e^\frac{-x^2}{2α^2}, x ≥ 0, $ where α is the scale ...
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Is there an informative term for calling the random elements conditional on which a PDF of a random element is defined?

Let $X_{1}, \dots, X_{n}$ be i.i.d. random elements; suppose the conditional PDF $f_{X_{1} \mid X_{2} , \dots, X_{n}}$ exists. Then I wonder if there is already in literature an informative name for $...
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Determine the joint pdf of two new variables

I'm working on the following problem: and here's my attempt at a solution: From what I understand, $0<=x<=y<=1$ can be separated into $0<=x<=y$ and $x<=y<=1$, so when I plug in ...
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What is the probability density function of N(x;…) * N(x;…)?

Task: What is the PDF of $$ p(x) = \mathcal{N}(x;\mu_1, \sigma_1)\mathcal{N}(x;\mu_2, \sigma_2) $$ Hint: what distribution will the PDF belong to? Maybe you can simply compute the new mean and ...
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36 views

Compute Mean of Normal Distribution where x% of Values are over y

I am looking for a way to determine the mean of a normal distribution (with given variance), where e.g. $z = 0,37 = 37\% $ of values should be above a certain value $a$ (e.g. 0,2)? My first idea was ...
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58 views

Is there a skewed probability density function that models a normal distribution with two parameters, $σ_1$ and $σ_2$?

Is there a way to model data that are skew normally distributed, but for which one builds in two seperate standard deviations? The parameter $σ_1$ should specify the 15.9% to 50% interval, whereas $...
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Variance being negative

Let $X$ and $Y$ have joint pdf such that $$f(x,y) = 3e^{-3x-y}, 0 < x< \infty, 0< y< \infty.$$ (a) Show that $X$ and $Y$ are independent. (b) Calculuate $Var(X)$. In ...
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1answer
31 views

What does PDF of normal distribution represents? [duplicate]

I have a basic question about the probability density function of the standard normal distribution $X\sim N(0,1)$. I understand that the cumulative distribution function for x is $P(X\le x)$ (in R ...
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Is the parameter vector of an indentifiable distribution of a transformed random vector always a subvector…?

I would like, after further considerations about this problem, to reformulate this question of mine again. I kept a record of the past words and remarks as the appendix below. I think the question ...
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How to estimate distribution given min-max intervals

I have a range of interval data, with start and end being considered as (kind of) minimum and maximum values for an unknown parameter (or perhaps they can be viewed as some estimates of wide (say 99%) ...
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1answer
56 views

Probability of first time to an event

We have a stream of events over time. Suppose that $f_t$ is the probability density that an event happens at time $t$. For example, $f_t$ can be the probability density that any bus arrives at time $t$...
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Joint pdf from joint cdf in R

I have a big matrix of data where for each element (column) I have a certain number of values. Using this data I computed, using the Emcdf library the Empirical Joint CDF for every couple of elements. ...
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39 views

Applying assumptions about marginal and conditional PDFs

We are given $0 < x_2 < x_1 < 1$. What assumptions can you make about $f_1(x_1)$ and $f_{2|1}(x_2|x_1)$? I know that $f(x_1) f_{2|1}(x_2|x_1) = \frac{1}{x_1}$. I know the expression can be ...
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How to interpret the probability density function exceeding one over a finite interval? [duplicate]

If one looks up 'Frechet Distribution' on wikipedia, one will find the following figure in the top-right of the page I was under the impression that the integral of the PDF function taken from ...
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Is it possible to find the joint distribution of a random vector if only the distribution of scalar many-to-one transformation is known? [duplicate]

Theoretical Exercise: I'd like to derive the joint distribution $p_{\boldsymbol{X}}$ of a random vector $\boldsymbol{X} \in \mathbb{R}^K$ if only the distribution of a scalar many-to-one ...
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1answer
22 views

How to scale between a equal distribution and an empirical distribution

I am not that good at expressing things mathematically, so I'll start with the practical problem right away: I have a set of four objects: O1, O2, O3, O4. Now I want to assign a variable that scales ...
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How to integrate probability density of sum of two indepedent random variables with a finite lower bound on one of them?

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-u^2/2}\:du=1$$ but $$u = \ln(A)-C-k$$ where $\ln(A)$ and $C$ are normally distributed independent random variables, and $k$ is a constant. I am ...
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1answer
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How many times must I roll a die to confidently assess its fairness?

(Apologies in advance for use of lay language rather than statistical language.) If I want to measure the odds of rolling each side of a specific physical six-sided die to within about +/- 2% with a ...
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3answers
103 views

what does p( y | μ,σ²) really mean?

Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically: I understand what p( A | B ) where A="I am sick" and ...
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multivariate normal distribution range [duplicate]

Simple question about MVN pdf. I understand the domain to be [0,1]. However, why does scipy.stats.multivariate_normal.pdf output values above this range. E.g. <...
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62 views

Distribution of maximum frequency of uniformly distributed integers

If I roll an M sided dice N times, there will be at least one number that occurs most frequently. What's the distribution of that maximum frequency in terms of M and N? (its pmf and name if it has one)...
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Finding probability of a point using bivariate copula density

I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
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What is “data distribution”? Does it Belong to Probability Space?

While reading the paper BEGAN : Boundary Equilibrium Generative Adversarial Network the autor writes as following: "the generator $G(z)$, which maps a sample $z$ from uniform distirubiton to the data ...
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Computational complexity of sampling from discrete and continuous distributions?

What is the computational complexity of sampling from any of these cases? I mean the computational complexity of the most efficient existing algorithm, not a possible algorithm or a lower bound. ...
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Drawing density plot in R [duplicate]

I drew a histogram of my data: ...
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1answer
29 views

Generate PDF from CTMC

I have an irreducible continuous-time Markov chain (CTMC) with a finite state space. The CTMC also does not have any one-step transitions from any state to itself. I have the transition rate matrix $Q$...
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1answer
52 views

how to get joint pdf of mixed random variables

I would like to know how the joint probability density function $p(b,r,\sigma^2)$ can be calculated for the following graph. Random variable $b$ is a latent binary variable, and random variable $\...
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1answer
84 views

Statsmodels' Negative Binomial: after .fit_regularized(), how to turn PMF into PPF to get the discrete values?

I used the package statsmodels to fit a Negative Binomial to my data. This data contains ~1500 samples with 21 covariates. Since I have overdispersion in my data ...
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24 views

More general conditional PDFs

The definition of the conditional PDF of X, given Y=y, is well known. How would one define the conditional PDF of X, given X>A (which I believe is called the Hazard Function), in a formal manner? I ...
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1answer
20 views

pmf for coin toss

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up. Let $x=0$ represent a 'heads' outcome and $x=1$ represent a 'tails' ...
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34 views

How do I compute these values from this PDF?

I can not figure out how to find $P(1.9\leq|X|\leq3)$ when $1.9$ is not given as any value of $x$. Would that just be the $P(X=3?)$ I did what I knew how to do, but I am not sure how to proceed from ...
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How can I find this constant?

My friend asked this question in our class: let X be a random variable which has a cumulative distribution function . Find (a). I think (a) cannot be solved but my other friend thinks (a) = 5/8 ...