Questions tagged [density-function]

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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Is there a meaning to the integral of $x \times f(x)$ over a range that is not infinite?

I know that the expected value can be computed as : $\mathbb{E}(X) = \int_{-\infty}^{\infty}xf(x)dx$ What if we do not do the integral over the whole range but only up to some value? Would there be ...
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How to prove symmetry of a Uniform kernel?

I am trying to prove this kernel is valid, $$ K(x) = \frac{1}{2}I(-1 < x < 1) $$ So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$ Also, how do we satisfy that k(x) is $\ge$ 0 for ...
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Question Evaluating PDF of Transformed R.V

I'm curious if the following method is valid, or if I need to use the gradient of the derivative as is typical when computing the pdf of a transformed r.v: Let $g(\theta) \sim f$, where $g(\cdot)$ is ...
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2 answers
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How to go from this formula to F distribution

I have the following formula. $$\int_{0}^{\infty} x^{\frac{n-1}{2}-1} (a+x)^{\frac{1}{2} \frac{-n}{2} -v} dx.$$ The quantities $d1 ,d2$ appearing in the pdf of the F distribution are the following. $$...
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Calculate the likelihood at the mode of a pdf conditioned on θ [closed]

Assuming that I have the PDF of a random variable X with parameter θ (i.e., f(x|θ)). How is the likelihood at the mode of this PDF computed?
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Estimate at which point a linear model hits a certain value (including probabilities)

I have a simple 1D set of datapoints with a trend, I want to estimate at which point $X_t$ (i.e., at which point in the future) the model will hit a certain threshold $Y_t$: I can fit a trendline to ...
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Estimated lognormal PDF shifted compared to data

I have experimental data (~100k observations) that appear to be from a lognormal distribution. I am attempting to estimate the distribution parameters using ...
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2 votes
1 answer
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The probability/cumulative density function for inequality of two random variables

I have two random variables X and Y which came from different inverse gaussian (IG) ...
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Find the MLE density function of uniform [-\theta,\theta] [duplicate]

For $X_1,\dots,X_n$, i.i.d $X_n \sim \mathrm{unif}[-\theta,\theta]$, the ML: $\hat\theta_{MLE}=\mathrm{max}\{-X_{(1)},X_{(n)}\}$. Find the density function. Hint: For $x_1,\dots,x_n$ : $\textrm{max}\{-...
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What is the probability mass function of Rock, Paper, Scissors?

I was curious about the statistics behind the game of rock, paper, scissors. Let's say n people are playing, where n is greater than or equal to 2. If when all n people reveal their play and only 1 ...
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Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability?

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The distribution function of appropriately normalised trimmed mean $T_n(\hat{F}_n)$

A definiton of functional of trimmed mean: $$ T^{\alpha}_n = (n-\lfloor \alpha n \rfloor)^{-1} \sum_{\lfloor\alpha n\rfloor + 1}^{n-\lfloor \alpha n \rfloor} X_{(i)}$$ A definition of the optimal ...
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Question on solution of Casella and Berger Exercise 9.10: Showing that $Q(t,\theta)$ is a pivot

My question concerns Exercise 9.10 of Statistical Inference by Casella and Berger: On page 428 the authors say In general, suppose the pdf of a statistic $T$, $f(t|\theta)$, can be expressed in the ...
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1 answer
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How is the q(z) function added at the end of this Bayesian formula?

At the bottom of this Bayesian formula why is a q(z) is brought into numerator and denominator positions? Is this within the rules of Algebra? Could anything be placed in the numerator and denominator?...
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Issue about both results in agreement with 2 different ways to compute variance of a random variable : weighted chisquared vs Gamma distributions

1.) I am interested in computing the variance of this observable $O$ involving the coefficients of spherical harmonics $a_{\ell m}$ and the $C_{\ell}$ which is the variance of an $a_{\ell m}$ : $$O=\...
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Get probability distribution function from density function and calculate the cumulative value [duplicate]

For the given density function, how to find its distribution function and how to calculate the value of the distribution function? Density function: $$f(x) = \frac{1}{\Gamma(\frac{n}{2})}x^{\frac{n}{2}...
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Estimate 2D covariance from 3D matrix where 3rd column contains probability density values

I have an nx3 matrix where the first 2 columns contain uniformly distributed random (x, y) points and the 3rd column contains pdf values evaluated at each point. The pdf values are computed from a ...
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Kernel Density: How do the terms 'global' and 'pilot' translate?

I nearly most of the articles on kernel smoothing or concepts that use kernel density estimations, authors speak of 'pilot' and 'global'. https://link.springer.com/article/10.1023/A:1008925425102 &...
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Pdf of a function of a complex random vector

Assume the complex random vector $\mathbf{y} \in \mathbb{C}^N$ is distributed as $\mathbf{y} \sim \mathcal{CN}(\mathbf{\mu},\mathbf{R})$ $$ p_{\mathbf{y}}(\mathbf{y})=\pi^{-N}\textrm{det}(\mathbf{R})^...
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Probability density definition in Jaynes Probability Theory

I've a question for those who have read the book Probability Theory of Jaynes. In the paragraph 4.5, where the author introduces probability density functions for the first time, I don't understand ...
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Computing Gini coefficient for a 2 parameters density function

I have a random variable $X$ defined by the following the density function, \begin{equation} f_{\theta_1, \theta_2}(x) = \begin{cases} \frac{\theta_1 \theta_2^{\theta_1}}{x^{\theta_1 + 1}}, &...
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What is the pdf of the multiplication of two normal random variables? [duplicate]

I want to know the pdf of the multiplication of two normal random variables (may or may not have the same mu and sigma, may or may not be correlated). ...
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Similarity measures for Probability Mass Functions

I am trying to predict whether two sets of papers are written by the same author by looking at the distribution of papers over the years (number of papers published in a given year). Suppose we have ...
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Determine expected goals from two normal distributions

Assume Team A scores an average of 3 goals per game with a standard deviation of 1.0, and assuming Team B allows an average of 2 goals per game with a standard deviation of 0.5. How would you ...
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3 votes
1 answer
329 views

Can non-parametric data have mean value and standard deviation?

I understand that for non-parametric data, the probability density function (pdf) cannot be obtained using parameters like (mean value) and (standard deviation), and I understand that we use Kernel ...
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PDF of the exp-gamma distribution

exp-gamma distribution is defined as the density of the random variable log(X) when X is a gamma random variable. I am trying to obtain its PDF. Unfortunaltely, the only formula I have found is from a ...
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Density function of nonlinear combination of normal random variables

Say we have two random variables $A,B \sim \mathcal{N}(0,1)$ and they form the following combination $$ X = A^2 + B^2 - \frac{A^2 B^2}{A^2 + B^2}. $$ Is there any way to obtain the probability ...
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3 votes
1 answer
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Is this statement of a stationary density function correct?

I'm planning to use a discrete-time stochastic process defined in the following paper: Nicolau, J. (2002). Stationary Processes That Look Like Random Walks—The Bounded Random Walk Process in Discrete ...
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pdf of sum of convex combination of two random variables [duplicate]

This paper claims: If we have two random variables ξ1 and ξ2, then we can form their mixture if we take ξ1 with some probability w and ξ2 with the remaining probability 1 − w. The probability density ...
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Compute P(XY) given P(x) and P(y)

Given two random variables x and y. Their PDFs P(x) and P(y) are known. However, if we do not assume the independence between x and y, how can we represent the Cumulative Distribution Function F(xy) (...
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2 answers
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Histogram of a Sample with Overlay of Population Density

To familiarize myself with histograms and probability density functions, I decided to sample various distributions, plot samples' histograms and their corresponding probability distribution functions. ...
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2 answers
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create probability function of my dataset

I want to create a probability density function of my dataset. I follow the advice of a specialist to visualize my dataset...Which is distribution fits my dataset visualization? Thank you!!! Your ...
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6 votes
1 answer
249 views

How to find the PMF of a weighted sum of IID Bernoulli random variables with constant sum of weights

Let $\{X_1,X_2,\ldots X_k\}$ denote a set of $k$ IID $Bern(p)$ random variables. Also, I have a set of $k$ non-negative integer weights denoted by $\{a_1,a_2,\ldots a_k\}$ such that $\sum_i {a_i}=k$. ...
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Problem on Discrete Random Variable

Please kindly give a pointer to this question. Generating the discrete variables seems unlikely! Entrance to a country can be denied for a number of reasons. When someone arrives by air, and their ...
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Fisher Information of Weight in Mixture distribution

Let's assume $x$ follows a mixture of two arbitrary continuous probability distributions with probability density functions $p_1(x)$ and $p_2(x)$, respectively. The probability density function of $x$ ...
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1 answer
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Distribution Function from Density Function

I'm guessing there was an error in a Probability and Statistics exam I have recently taken. Let $X$ be a random continuous variable and $f$ a function defined as follows: $ f(x)=\left\{\begin{matrix} ...
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Joint density and univariate time series models

Is the picture below an example of univariate time series model since I am observing the same random variable just at different time points? Can joint density be used to explain univariate time series ...
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Finding probability of all success for in an order statistic

𝑓(𝑦) = 5𝑦^4; 0 ≤ 𝑦 ≤ 1 A group of 3 friends order small cups of soda, from the soda dispenser. If the 3 small cups are considered a random sample from the dispenser fills, find the probability ...
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Method for selecting points from a dataset with known distribution A, so the selected points have known distribution B

I have a dataset (10K points) that was sampled from distribution A (the pdf of distribution A is known). I want to select a subset of the data (1k points), so the selected points have distribution B (...
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Density function of a dependent sum of products of normal random variables

Say we have a random variable $$ X = A_0 A_1 + A_0 A_2 + A_1 A_2, $$ which consists of normally distributed independent random variables $A_0, A_1, A_2 \sim \mathcal{N}(0,1)$ with probability ...
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Expected distance under Gaussian noise

Summary I'm working on a tracking problem, where I'm trying to estimate the position of an object that moves in on plane. In my simulator, at each sampling step I generate a measurement that is given ...
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1 answer
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Random numbers with exponentiated gamma distribution? [closed]

How to get random numbers following "exponentiated gamma distribution"? I tried to search some functions in R and this is what i got: https://rdrr.io/cran/Newdistns/man/expg.html I want to ...
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Why pdf is not considered and cdf is considered for estimating probability of a value [duplicate]

I am a learning statistics on my own and I am a beginner. I came to know that to construct a distribution curve one has to first create histogram. Let's assume that it is a normal distribution. Then ...
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Gaussian closed form solution of marginal likelihood

i have tried some time now to understand a specific step in the derivation of what I think is a marginalization integral. I am still learning about these statistical things and I think I miss ...
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Finding density function of random vector (density transformation)

Let $X_{i} \sim \operatorname{Ga}\left(\alpha_{i}, \lambda\right)$ independent with $\alpha_{i}>0$ fūr $i=1,2$. Furthermore it is known, that $V=X_{1}+X_{2} \sim \operatorname{Ga}\left(\alpha_{1}+\...
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How to show $X \text{~Uniform}[-1,1] $ and $Y=-X$ when $X\leq 0$, $Y=X$ when $X \geq 0$, then $Y \text{~Uniform}[0,1] $?

Told to show that: if $X \text{~Uniform}[-1,1] $ and $Y=-X$ when $X\leq 0$, $Y=X$ when $X \geq 0$, then $Y \text{~Uniform}[0,1] $. [where X,Y are continuous random variables] I can see why it holds ...
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2 votes
1 answer
58 views

What is the difference between a probability measure and a probability density function? [duplicate]

During my research, I have repeatedly come across the terms probability measure and probability density function (pdf). I am familiar with the concept of a pdf, but I am not entirely sure how ...
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4 votes
2 answers
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Show that a Linear transformation does not change shape of a distribution

It is easy to show how a linear transformation affects the mean or the variance of a distribution. It is easy to find over the internet that a linear transformation does not change the shape of a ...
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When does a group specific dispersion parameter for the negative binomial distribution make sense?

If you have overdispersed observed abundance of multiple species including zero inflation the negative binomial distribution seems to be a reasonable choice. But if some species occur much more ...
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2 answers
96 views

Do We Need CDF, When We Have A PDF? [duplicate]

CDF is the probability that a random variable takes on a value less than or equal to a fixed $x = a$. Assuming we have a a random variable $X$ that has a PDF, both CDF and PDF have the same ...
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