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:

$$Pr(a \le X \le b) = \int_a^b f_X(x) \, dx$$

2011 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…
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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?
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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…
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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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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…
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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} =…
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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…
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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 $F'$ in continious…
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Non standard solution to $f(x) = \frac{1}{2}\Big(f(\frac{x}{2}) + f(\frac{1+x}{2})\Big)$

Final update: on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here. This functional equation appears in the following context. Let $\alpha\in[0,1]$ be an irrational number…
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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…
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Are all probability density functions described by their mean and variance?

The question might be trivial, but I would like someone to correct me or confirm it. I know that the Normal (Gaussian) distribution is completely determined by its mean and variance, but does that hold for any other distribution? I assume that the…
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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…
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Pdf of sum of exponential random variables

I am trying to understand the following: Let $X$ and $Y$ be independent and exponential random variables with parameter $1$, so $f_X(x) =e^{-x} $ and $f_Y(y) =e^{-y} $. Let $Z=X+Y, X
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Unique question about packing problem

I added the related pages from part 3 of the book: combinatorial geometry by János Pach,Pankaj K.Agarwal (1995) (which is not available on net so I added them as pictures). A. Prove that one can always find a packing $Ç$ of the plane with congruent…
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