Mathematics Stack Exchange
Mathematics Stack Exchange

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:

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

2263 questions
3
votes
1 answer

density of set question. help appreciated again!

Here's another question I don't know what to do with. Compute the Shnirelman density of the set $$A = \{ \lfloor n \ln (n) \rfloor : n = 1,2,3,... \}.$$ I know that the Shnirelman density of a set $A$ is $$\inf \left\{ \frac{ | A \cap…
Senhor José
  • 69
2
votes
1 answer

support of density function

A random variable Y has density function f(y) = y/4 for 1<=y<3, and 0 elswhere. The question states: Find the support of U=3Y -2, that is , find the range of real numbers u for which the density function of U is positive. I am not sure what is the…
Denson
  • 91
1
vote
0 answers

Add Skew to Distribution without Changing mean or variance

How can I change the Skew of a PDF without changing the Mean or Variance? And more generally, how can I change the Nth moment without changing any of the lower moments? Vielen Dank im Voraus !
TPM
  • 111
1
vote
0 answers

The probability density of a function of a random variable

I would like to know how my professor had these two results: First: Let X be a r.v uniformly distributed in [0, 2\pi], what's the density function of $Y = \sin(X)$. He gave as an answer $$\frac{1}{\pi\sqrt{1-y^2}}$$ but I would say that the answer…
Ino
  • 11
1
vote
1 answer

Are densities square integrable?

I am reading a paper which deals with the densities of $(X_t)$ where $X_t$ is typically a Levy process. The paper assumes that the density of, say, $X_1$, is square-integrable, i.e. in $L^2$. Other times, it will assume that the characteristic…
saei
  • 77
1
vote
1 answer

Density argument

This is a problem that I met when studying density things: Let $\{ a_n \}$ be a real-valued sequence. Then the following things are equivalent: (1) $\lim_{n \rightarrow \infty } \frac{1}{n}\sum_{i=1}^{n}a_n=0$ (2) There exists some $J \subset…
lmksdfa
  • 335
1
vote
3 answers

Density of connecting lines inside a circle

A circle can be seen as an infinite number of points which are all placed at the same distance from the center. Let's start with a polygon of $n$ points. The polygon approaches a circle for $n \rightarrow \infty$. Now lets connect every corner point…
Gilfoyle
  • 401
1
vote
1 answer

meaning of ratio of density function to distribution function of the standard normal?

Consider the model: $ y_i = \beta' X_i + u_i $ if RHS $> 0.$ $ y_i = 0$ otherwise. So the expected value of $y$ under condition of $y_i > 0$ is: $$ E(y_i| y_i >0) = \beta' X_i + E(u_i|u_i > -\beta'X_i)$$ $ E(y_i| y_i >0)= \beta' X_i + \sigma…
Vũ Võ
  • 113
  • 5
1
vote
1 answer

Let $X$ and $Y$ be independent random variables with densities $f_X(x)=\cdots$

Let $X$ and $Y$ be independent random variables with densities $$f_{X}(x)= \begin{cases} \gamma e^{-\gamma x} & \text{, } x\geq0 \\ 0& \text{, } x<0 \end{cases}$$ $$f_Y(y)= \begin{cases} \mu e^{-\mu x} & \text{, } y\geq0 \\ 0& \text{, }…
beepboopbeepboop
  • 223
1
vote
1 answer

Standardization of density functions

I have the density function $$f(x)= \frac {e^x}{(e^x+1)^2}.$$ The integral from $-\infty$ to $\infty$ is $1$ so it is indeed a density function. The expected value of the function is $0$ and the variance is $\pi^2/3$. My goal is to set the variance…
user2968163
  • 23
1
vote
1 answer

Simultaneous density function of $n$ $\operatorname{hom}[0,\theta]$ distributed variables

Say we have $X_1,\dots,X_n$ independent random variables that are $\operatorname{hom}[0,\theta]$ distributed with parameter $\theta>0$. I don’t understand why we have $$ p_\theta(x_1,\dots,x_n)=\prod_{i=1}^n\frac{1}{\theta}\mathbb 1_{\{0\leq x_i\leq…
Sha Vuklia
  • 3,960
  • 4
  • 19
  • 37
1
vote
0 answers

Meaning behind normalising a density function

What does it really mean to normalise a function such that it becomes a probability density function? Say we have a nonnegative integrable function $f:\mathbb R\to\mathbb R$. The normalising procedure would be as follows: $$ \int_{-\infty}^\infty…
Sha Vuklia
  • 3,960
  • 4
  • 19
  • 37
0
votes
0 answers

Density results in L2 space and H1

i read in Brezis book that 1)the spaces $C_{0}^{\infty}(\omega)$ and $C^{\infty}(\omega)\cap H^{1}(\omega)$are dense in $H^{1}(\omega)$ so can we say that also $C_{0}^{\infty}(\overline{\omega})\cap H^{1}(\omega)$ is dense in $H^{1}(\omega)$?…
Amira
  • 11
0
votes
0 answers

Joint density of discrete and continuous variables.

Let $X$ and $Z$ be discrete variables, and $Y$ is a continuous variable, I am interested in the joint density $g(x,y)$, and if the following equality holds considering any assumptions? $\sum [P(x)g(y|\theta_x)]=\sum [P(z) g(y|\theta_z)]$
Md. Ashraful Islam
  • 1
0
votes
1 answer

Proof that joint density of two rv is equal to the conditional density of first given the second times the second

I would like to know why this equality $f_x,_y (x, y; θ) = f_xy (x; θ) × f_y(y; θ)$ holds. so, why the joint density of two random variables is equal to the conditional density of the first given the second times the second? It's a given axiom in…
MaxSim423
  • 1
1
2 3