# 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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### 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, 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 ...