Questions tagged [density-function]

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

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Distributions over a unit square

Problem definition Consider the following bivariate random variable \begin{equation*} \overline{z}\triangleq \frac{1}{4} \sum_{j=1}^4 z_j \end{equation*} where $z_1, \dots, z_4$ are uniformly ...
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Why is absolute convergence required?

Definitions try to precisely formalize intuitive ideas. For example, the concept of mathematical expectation attempts to rigorously formalize the idea of quantifying the average outcome of a random ...
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Probability density function of three independent random variables

I will try to describe the question first and how i solve the problem. Here we have three independent random variable $X,Y,Z$ and $S_3=X+Y+Z$, the probability density function of three variable are ...
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New variable with same density under change of measure

If $X$ is a random variable defined on $\mathbb{R}^n$ endowed with the Borel sigma-algebra and the Lebesgue measure $\lambda$. X has a density, $f$. For $k>0$, consider the density $$g(x_1,\ldots,...
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What I am doing wrong in this transformation of random variables

Let $X_1$ and $X_2$ be independent standard normal variables. Find the joint density function of $Y_1 = X_1^2 + X_2^2,\quad Y_2 = \frac{X_1}{X_2}$. My solution: After solving I have $\frac{1}{|J|} = 2\...
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Can anyone answer Part C and explain to me if I am doing correctly? [closed]

I'm attempting to solve Part C which I know involves comparing the product of the marginal of x and y to f(x|y). If they are the same then they are independent. I got the marginal of y as 1/(2sqrt(pi))...
Guhan Gnanam's user avatar
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Which is the domain of the probability density function of $X^2$?

Let $X$ be a random variable with probability density function $$f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \text{ for each } x \in \mathbb{R}.$$ We want to find the probability density function ...
Cyclotomic Manolo's user avatar
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Find the marginal density function using integration by substitution

Consider the function $f(x_1,x_2,x_3) = \frac{1}{\sqrt{2\pi^3}}*\Bigl(e^{\frac{1}{2}(x_1^2+x_2^2+x_3^2)}\Bigr)*\Bigl(1+x_1x_2x_3*e^{\frac{1}{2}(x_1^2+x_2^2+x_3^2)}\Bigr)$. I am trying to find its ...
l337n00b's user avatar
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"Deconvoluting" the sum of independent random variables

I encountered the following question in my research. See here for some background, but this post should be self-contained. Suppose $X$ follows a discrete uniform distribution on $m$ points $\{X_1, ...
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Integral of a function involving tangent and its exponential [closed]

How could we find the close form of this integral? \begin{align} f_X(x) = \int_{0}^{\pi/2} \tan^{\alpha+1}(\phi) e^{-\tan(\phi)(x+\beta)} d\phi \end{align}
arnawa's user avatar
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How to prove that the density function integrates to 1 over the whole support?

Show that the function $f(x,y)$ when the function is given by: $$ f(x,y) = \begin{cases} \frac{15}{2}x^2y &\text{for } -1<x<1,|x|<y<1, \\ 0 &\text{else}. \end{cases} $$ I could ...
Selby Ching's user avatar
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How does this density function make sense?

Let $X$ be an exponentially distributed random variable with parameter $\beta>0$ and let $Y$ be a random variable with $P(Y=0)=1-p$ and $P(Y=1)=p$, where $0<p<1$. Assume that $X$ and $Y$ are ...
Selby Ching's user avatar
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How to write the following conditional density function?

Suppose $X_1,X_2,X_3$ are independently and identically distributed with probability density function $f(x)$. Let event $A$ be "at least one random variable in $X_1,X_2,X_3$ is greater than $t$&...
ExcitedSnail's user avatar
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Predictive distribution for gamma likelihood and prior

For the following problem, I’m stuck at the 3rd question. I would appreciate if you could validate my answer to the 1st question as well. Problem Let $X_1, \dots, X_n \sim \operatorname{Gamma}(2, \...
Alexandre Huat's user avatar
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Would someone check my work using cdf method to find the density function?

Problem: Let Y be a random variable with pdf $$ f_{y}(y) = 2(1-y) \quad 0 < y<1$$ Using the cdf method, find the density function of $U_1 = 2Y-1$. This seems straightforward, but I would be ...
Calum's user avatar
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Finding the joint density function of $(X+Y,X)$ with $X$ and $Y$ independent and following an exponential distribution with parameter $\lambda>0$

Given two independent random variables $X$ and $Y$ that both follow an exponential distribution of parameter $\lambda > 0$, I am trying to find the joint density function of $(X+Y,X)$. I have ...
Gabriel Gontier's user avatar
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Understanding the following expression related to independent random variables

I am having the following expression: $$P = \text{Pr}\biggl(h_0 h_1 < \frac{u_1}{a_1 \rho - a_2 \rho u_1}\biggr)\tag{1}$$ where $h_0,h_1$ are independent and identically distributed (i.i.d) random ...
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Understanding the following expression related to random variable

I am having the following expression: $$P = \text{Pr}\biggl(h_0 h_1 < \frac{u_1}{a_1 \rho - a_2 \rho u_1}\biggr)\tag{1}$$ where $h_0,h_1$ are independent and identically distributed (i.i.d) random ...
Heretolearn's user avatar
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Distribution Function Symbol Problem

i am doing my exercises about distribution and this is the density function. tho i have never seen in my lecture the $\mathbf{1}_{[a, b]}(t)$ symbol and i have no one to ask about it. i dont ...
haminta kurti's user avatar
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Integral involving the CDF of a normal and a nonlinear function

I know that the following result holds Given $a,b,\lambda\in \mathbb{R}$, $\Phi$ the cumulative distribution function of a standard normal random variable, and $f_X$ the probability density function ...
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How do you determine the amount of mass for both a solid and diffuse object?

Given the value $K$ multiplied by radius $r$, I can determine the amount of mass of the solid red sphere at the center of the image using the following: $K * r = mass$. However, $K$ begins to diminish ...
Tivity's user avatar
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Formula related with the partition theorem

I am stuck with solving this problem: Let $X$ be a continuous random variable with probability density function $f_X$. Then, $$ f_X(x)=f_{X\mid X<0}(x)\cdot P(X<0)+f_{X\mid X\ge 0}(x)\cdot P(X\...
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Understanding the following CDF expression

I am trying to understand how the following CDF expression is obtained. $h_1 = \text{min}(f, g)$ ---(1) where $f, g$ are independent and identically distributed random variables. $F_{h_1}(h) = \text{...
Heretolearn's user avatar
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When is $\overline{d}(A \cap A/2)>0$?

Let $A \subset \mathbb{N}$ have $\overline{d}(A):= \limsup_{N\to \infty} \frac{|A\cap \{1,\ldots,N\}|}{N} >2/3$. Is it true that $\overline{d}(A \cap A/2)>0$? What I can show is that this holds ...
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Analyzing Point Distributions in Voronoi Tessellations from Two Probability Sources $p$ and $\tilde{p}$

Let's suppose I have a probability distribution $p$ and another distribution $\tilde{p}$. Suppose I sample $K$ points from the distribution $p$ which will be my centroids for my Voronoi tessellation. ...
Jose Manuel de Frutos's user avatar
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Mass of solid in 3 dimensions [closed]

Find the mass of the right pyramid that has a square base in the $xy$-plane. $-1<x<1, -1<y<1$, the vertex at $(0,0,8)$ and density function $f(x,y) = x^2$
Ghouliani's user avatar
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Find the density of random variable $X+Y$ for $f(x,y) = 6(x-y)$ if $0 \leq y \leq x \leq 1$

Find the density of random variable $X+Y$ for $f(x,y) = 6(x-y)$ if $0 \leq y \leq x \leq 1$. I am able to use method of transformation to convert $f(x,y) = 6(x-y)$ if $0 \leq y \leq x \leq 1$, and ...
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Probability density function satisfying $\lim_{x\to \infty} xf(x) = \lim_{x\to -\infty} xf(x) $

Let $X$ be a real-valued random variable with probability density function $f(x)$ (i.e., $f(x) \geq 0, \ \forall x \in \mathbb{R}$ and $\int_\mathbb{R}f(x)dx = 1$), which is also assumed to be twice ...
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Prove $\frac{e^{t_xX_1}}{\varphi(t_x)}$ is a probability density function

Let $X_1$ be a random variable. I need to prove that the function $f(x_1) = \frac{e^{t_xX_1}}{\varphi(t_x)}$ is a density function, whereby $\varphi(t)$ is the moment generating function of $X_1$ and $...
user996159's user avatar
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How to find the probability density function of a random variable that is related to another random variable?

I found a similar question already answered here: Find CDF of random variable which depends on other variable Our goal is to find $f_Y$ (the PDF) which is the derivative of $F_Y$ (the CDF). $X$ is a ...
McSuperbX1's user avatar
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Tsiatis Semiparametric Theory Chapter 2 Exercise 3(b) Use Change of Variable to Get a NEW Probability Density

I have been reading the book Semiparametric Theory and Missing Data by Tsiatis. The question is about exercise 3(b) in chapter 2 Hilbert Space for Random Vectors. Here is the problem Let $Z=\left(Z_1, ...
maskeran's user avatar
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Exchange differentiation with integration: $E[f'(X)/f(X)X]$

Let $X$ be a random variable with p.d.f. $f(x)$ (absolutely continuous and differentiable everywhere) and let $s(x)$ be the density score, i.e., $s(x) = d\log f(x)/ d x = f'(x)/f(x)$. Assume $\int |x| ...
swan11's user avatar
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$\phi(t)=\sqrt{1-t^2} \ $ if $|t|<1$ and $\phi(t)=0$ if $|t| \geq 1.$ Prove that $\phi(t)$ is not a characteristic function

$\phi(t)=\sqrt{1-t^2} \ $ if $|t|<1$ and $\phi(t)=0$ if $|t| \geq 1.$ Prove that $\phi(t)$ is not a characteristic function. $\phi(t)$ checks obvious signs of a characteristic function ($\phi(0)=1,...
fragileradius's user avatar
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Conditional probability density functon is a probability density function

How do you formulate the proof of "conditional probability density function $f_{X|Y}(x|y):=\frac{f(x,y)}{f_Y(y)}$ ($f(x,y)$ is the joint probability density function, and $y$ is restricted to the ...
Iris's user avatar
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Is it meaningful to define a probability distribution (or measure) by projecting the uniform defined on an arbitrary curve?

I have learned that the common ways to define distributions (measures) are by either density probability functions or their integral, the cdf. I was wondering for fun if an alternate mechanism I am ...
Jada's user avatar
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Density function integrability problem

For random variables $A, Y, U$, they have density functions $p(a,u,y)$. Now I know $p(u|a)\in L^2(P_a)$ for $\forall u$, namely, $$ \int{\left( p\left( u|a \right) \right) ^2p\left( a \right) da}<\...
叶心萤's user avatar
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1 answer
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Equivalence of functions of random variables [closed]

I've come across the following claim regarding equivalence of functions of random variables, and I need help verifying whether this claim is true or not: We are given the following function of random ...
MRashid's user avatar
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Density of sum of two random variables

Let$(X,Y)$ be an RV of the continous type with PDF $f(x,y)$.Let $Z=X+Y$,then the Convolution of probability distributions told us the PDF of $Z$ is $f_{Z}(z)=\int_{-\infty}^{\infty}f(x,z-x)dx$. If we ...
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Find the joint density of a random vector

I consider $(T_1,…, T_n)$ a random vector where $T_i$ is initially a point process associated to $N_t$ the counting process. We know that $N_t$ is a time inhomogeneous Poisson process with intensity ...
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Density of normal c.d.f. of a normal random variable

Let $a$ be a constant, X is a standard normal and $\Phi$ is the c.d.f. of standard normal. what is the density of $Y$ where $$Y = \Phi(aX)$$ what is the density of $$Y=\Phi(X + a)$$ The second case ...
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About upper asymptotic density.

Let $A$ be a subset of $\mathbb{N}$ and $(p_i)_i$ a strictly increasing sequence of positive integers. We define $\overline{d}(A|(p_i)_i):=\limsup_n \frac{\left| A \cap \left\{ p_1,\cdots , p_n \right\...
johntree3's user avatar
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$E=\left \{ f(n)-\lfloor f(n) \rfloor ,n \in \mathbb{N} \right \} $

Let $f$ be a strictly increasing function such that: $\lim_{x\to+\infty} f(x)= +\infty$ and $\lim_{x\to+\infty} [f(x+1)-f(x)]= 0.$ Prove that $E=\left \{ f(n)-\lfloor f(n) \rfloor ,n \in \mathbb{...
ana nadi lwa3r's user avatar
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Need help calculating the density of $(a,b)X$ where $X$ has density $f$

I’m currently working on a problem where I need to calculate the density of $(a,b)X=(aX,bX)$ where $a,b>0$ and $X$ has density $f$ . However, I’m facing some difficulties as the Jacobian method, ...
user346624's user avatar
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The center of mass of a semiellipsoid

I am trying to find the center of mass of a semiellipsoid using cylindrical coordinates. $$ \frac{r^2}{a^2}+\frac{z^2}{b^2} \leq 1 $$ with $z < 0$ and density = 1. I know that the center of mass is ...
Mike Gotier's user avatar
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Single-crossing property for stochastic dominance

A useful criterion for first order stochastic dominance of two random variables $X$ and $Y$, denoted $X\le Y$, is to check whether the densities (or pmf if discrete) cross at most at one point. ...
xyz's user avatar
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Can Central Limit Theorem say anything about Probability Density Function?

For a sequence $\left( X_1, X_2, \dots \right)$ of independent and identically distributed random variables with zero expectation and unit deviation, CLT implies $$ \lim_{n\to\infty} P\left( \frac{1}{\...
Charlie's user avatar
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Can $\mathbb{P}[X>Y]$ be determined by looking at the area between the curves the curves $f_X$ and $f_Y$?

Say $X$ and $Y$ are 2 continuous random variables, where $f_X$ is the p.d.f of X, and $f_Y$ that of $Y$. I was wondering whether $\mathbb{P}[X>Y]$ is equivalent to the area between the curves $f_X$ ...
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Existence of a continuous function that satisfies a constraint: $\|f - f_{\epsilon}\|_{1} \leq \epsilon$ with $f(t) \in A$ a.e. $t \in [0,1]$.

Given a measurable function $f \in L^{\infty}([0,1],\mathbb{R}^d)$, we know that for all $\epsilon > 0$ there exists a continuous function $f_{\epsilon}$ such that $$ \|f - f_{\epsilon}\|_{L^1} \...
BrianTag's user avatar
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Let $(X, Y )$ be a random point in the triangle $\{(x, y) : 0 ≤ x ≤ y ≤ 1,x+y≤1\}$. Let $Z = \text{max}(X, Y)$. Compute the density function for $Z$. [closed]

Let $(X, Y )$ be a random point in the triangle $\{(x, y) : 0 ≤ x ≤ y ≤ 1,x+y≤1\}$. Let $Z = \text{max}(X, Y )$ be the larger of the two values. Compute the density function for $Z$. My work: $$\text{...
Christine's user avatar
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Joint Distribution Function of a Joint Density Function (exponential)

I have a density function $f(x,y)= 2e^{-x-y}$ for $0<x<y<\infty$ and $0$ elsewhere. What is the joint distribution function? So far I have calculated $F(x,y)= \int_0^x \int_0^yf(x',y') dy'dx' ...
gracetheangel's user avatar

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