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Questions tagged [density-function]

For questions on using, finding, or otherwise relating to probability density functions (PDFs)

A probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

A random variable $X$ has density function $f_X$, where $f_X$ is a non-negative Lebesgue-integrable function, if:

$$\operatorname{Pr}(a \le X \le b) = \int_a^b f_X(x) \, \mathrm dx.$$

2333 questions
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Density of sum of two independent uniform random variables on $[0,1]$

I am trying to understand an example from my textbook. Let's say $Z = X + Y$, where $X$ and $Y$ are independent uniform random variables with range $[0,1]$. Then the PDF is $$f(z) = \begin{cases} z & \text{for $0 < z < 1$} \\ 2-z & \text{for $1 \le…
Zhulu
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What is the difference between "probability density function" and "probability distribution function"?

Whats the difference between probability density function and probability distribution function?
Le Chifre
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How can a probability density function (pdf) be greater than $1$?

The PDF describes the probability of a random variable to take on a given value: $f(x)=P(X=x)$ My question is whether this value can become greater than $1$? Quote from wikipedia: "Unlike a probability, a probability density function can take on…
Newman
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Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function. Well, I know that $P(T_i>t)=e^{-\lambda…
TI Jones
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Is this distribution already known and has a name?

My question is whether the distribution on $\Bbb R$ with probability density $$ f(x) := \frac 2 {\sqrt{2\pi}} e^{-\frac{x^2}{2}} - 2 \vert x\vert \int_{\vert x \vert}^\infty \frac 1{\sqrt{2\pi}} e^{-\frac{y^2} 2} \text d y$$ is already appearing in…
Falrach
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Show that $\mathbb{E}\left|\hat{f_n}-f \right| \leq \frac{2}{n^{1/3}}$ where $\hat{f_n}$ is a density estimator for $f$

Question Suppose we have a continuous probability density $f : \mathbb{R} \to [0,\infty)$ such that $\text{sup}_{x \in \mathbb{R}}(\left|f(x)\right| + \left|f'(x)\right|) \leq 1. \;$ Define the density estimator: $$\hat{f_n} =…
yasinibrahim30
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convolution of $n$ exponential distributions

Let $exp(k)$ be the exponential distribution, $k>0$. Then it has density $$ f(x)= \begin{cases} ke^{-kx} & \text{ if } 0\leq x < \infty\\ 0 &\text{otherwise} \end{cases} $$ I want to find the convolution of $n$ exponential distributions. For…
Sarah
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Transformation of Random Variable $Y = X^2$

I'm learning probability, specifically transformations of random variables, and need help to understand the solution to the following exercise: Consider the continuous random variable $X$ with probability density function $$f(x) = \begin{cases}…
user347616
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Conditions for the existence of a density with respect to Lebesgue measure

Let $X:\Omega \to \mathbb{R}$ be a random variable on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and denote by $$\chi(\xi) := \mathbb{E}e^{i \xi \cdot X}, \xi \in \mathbb{R}^d,$$ its characteristic function. I'm looking for sufficient and…
saz
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Geometric Interpretation of Product of Two Multivariate Gaussians densities

I am trying to understand the high-dimensional geometry behind Bayesian estimation. When you multiply two Normal densities with respective means $\mu_1, \mu_2$ and covariances $\Sigma_1, \Sigma_2$, the renormalized product is again a Normal density…
Maximilian Sölch
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How to derive the density formula for $Y=h(X)$ when $h$ is not injective?

Let $X$ be a real-valued random variable with density $f_X$, and let $Y = h(X)$, where $h: \mathbb{R} \to \mathbb{R}$ is a continuously differentiable function. Assume that for every $y \in \mathbb{R}$, the preimage $h^{-1}(y)$ is at most…
mathematico
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PDF of $z=\frac{AB-CD}{A^2+B^2+C^2+D^2}$, where $A$, $B$, $C$, and $D$ are independent Gaussian random variables with mean $0$ and variance $1$

I would like to find the PDF of the real random variable $$z=\frac{AB-CD}{A^2+B^2+C^2+D^2},$$ where $A$, $B$, $C$, and $D$ are independent Gaussian random variables with mean $0$ and variance $1$. Alternatively, the problem can be seen as finding…
Matteo Nerini
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Does a random variable with differentiable distribution function have density?

Question: Suppose that $X: \Omega \to \mathbb{R}$ is a random variable and it's distribution function $F(x) = \mathbf{P}(\xi \le x)$ is differentiable for all $x$. Is it true that $F'(x)$ is density? What do I know: Remark 1. If $F'$ is continuous…
Botnakov N.
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4 answers

Can a probability density function not integrate to 1 over its support?

I don’t know much Probability Theory beyond the undergraduate level. I was trying to model a simple scenario with my family. What is the probability I will develop type 1 diabetes in the following years? I did some research on the internet, and it…
Melanzio
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Distribution of $Y = X \bmod 2\pi$ with $X$ being a Cauchy distribution

Let $X$ be a Cauchy distribution with parameter $\theta$, that is to say, its density function is: $$ f(x;\theta) = \frac{\theta}{\pi(x^2 + \theta^2)} $$ I'm asked to get the distribution of $Y$ where $Y = X \bmod 2\pi$. I've tackled this problem…
Raúl Blázquez Bullón
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