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Questions tagged [density-function]

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

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Finding the CDF of a Sum of I.I.D Continuous Random Variables

Let X$_1$ and X$_2$ be identically independent distributions(i.i.d) random variables with $$\Bbb P(X_i \le x) = 1-x^{-1/2}, \quad x \ge 1 \ \text{and} \ i = 1,2 $$ Find $\Bbb P(X_1 + X_2 \le x)$. I ...
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Reasonable choice of KDE BW when changing kernel type?

For some distributions and some widely used kernels, the "optimum" bandwidth (e. g. according to L2 density error) is known. However, when switching the kernel, does it make sense to keep the kernel ...
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1answer
22 views

Probability density function on uniform distribution

Let $R=\{(x_1,x_2) \in \mathbb{R}^2 ~|~ x_1^2 + x_2^2 \leq 1\}$ be the unit ball in $\mathbb{R}^2$. It was stated to me that $\rho$ is the uniform probability on the square $[-1,1]^2$ and that $\...
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Finding a sufficient statistic when density function is given

Let $\mathbf{X}=(X_1,X_2,...,X_n)^T$ be a simply sample of random variable $X$ whose distribution belongs to family $\mathcal{P}=\{ f(x; \lambda, \eta, \mu ), 0<\lambda, \eta <\infty, -\infty &...
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If $X, Y$ ~ $\exp{\alpha}$ determine the PDF of $\frac {X}{X+Y}$

Let $X,Y$ be independent random variables the are distributed $\exp{(\alpha)}$ whereby $\alpha > 0$ Determine the Distribution Density of $\frac {X}{X+Y}$ My idea: I believe it is too simple ...
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1answer
11 views

Are the following functions probability densities on the intervals given - is my solution correct?

I don't have answers to refer to, so could please someone check if the way I solved this problem is correct? I have an exam tomorrow and want to make sure that I've got it right. THank you in advance! ...
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23 views

Conditional distribution to function

I'm doing a course on finance and there is a step in my notes that says the following:- Let $X$ be a random variable with distribution $F$. For a given $u$, let $M$ have the following distribution: $...
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Calculating probability using joint density

I'm given the family $(X, Y)$ of random variables with join density: $$f^{X, Y}(x, y) := 2 \cdot e^{-(x+2y)} \cdot 1_{[0,\infty)}(x) \cdot 1_{[0, \infty)}(y)$$ and I'm required to calculate $P(X >...
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How to find the parameter $c$ of a joint density function.

The question is: "Let $X$ and $Y$ have the joint density: \begin{equation} f(x,y)=cx(y-x)e^{-y} \tag{1} \end{equation} with $0 \leq X \leq Y \leq \infty$ 1 . Find $c$. In order to find $c$ the ...
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2answers
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Probability Density Function - Is my solution correct? Need help with a limit of an integral

I have no answers to refer to, hence, would be great if someone could check up if my procedure to solve the following problem is correct. Also, I am struggling to solve for event B from part b) - any ...
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Probability density function. Find min(X,Y).

Let $X$ and $Y$ have the same probability distribution with denisty function : $f(t)=0$ for $t \le 1$ and $f(t)=\frac{1}{t^2}$ for $t >1$. Find the density function $Z=\min(X,Y)$. We know that $$...
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2answers
52 views

Marginal distribution can't seem to work

Given $$D = \{(x,y)\in R^2:0\leq y \leq 1-|x| \}$$ And joint distribution function: $$ f_{X,Y}(x,y) = \begin{cases} 1 & (x,y)\in D \\ 0 & else \end{cases} $$ The exercise itself doesn't ...
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Given the joint probability density function of the random variables $X$ and $Y$, determine $\textbf{P}(X < Y)$

The joint probability density function of the random variables $X$ and $Y$ is given by $$f_{X,Y}(x,y) = \begin{cases} e^{-x-y} & \text{for}\,\,(x,y)\in(0,+\infty)^{2}\\ 0 & \text{otherwise} \...
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2answers
60 views

Assume that $X$ and $Y$ have joint probability density function $f_{X,Y}$. Calculate the joint probability density function of $U = XY$ and $V = X/Y$

Assume that $X$ and $Y$ have the following joint probability density function $$f_{X,Y}(x,y) = \begin{cases} \displaystyle\frac{1}{x^{2}y^{2}} & \text{if}\,\,x\geq 1\,\,\text{and}\,\,y\geq 1\\\\ \,...
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1answer
32 views

Naive question about continuous random variable: solution verification

The probability density function of the random variable $X$ is given by $$f(x) = \begin{cases} a + bx^{2} & 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$$ If $\...
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2answers
29 views

Find PDF of Y from (constant) joint PDF

Given the following joint PDF: $$ f_{X,Y}(t,s) = \begin{cases} \frac{2}{3} & 0\leq t\leq 1, -1\leq s\leq t \\ 0 & otherwise \end{cases} $$ I need to find $f_y(s)$ So according to defintion: ...
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1answer
18 views

Find marginal density at a point [closed]

Given the following joint density function: \begin{equation} f (x,y) = \begin{cases} 2& \text{} 0 \le x \le 1-y, 0 \le y \le 1\\ 0 &\text{otherwise} \end{cases} \end{equation} Find $f_{y}(...
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Given two independent identically distributed random variables, determine the distributions of $V = \max\{X,Y\}$ and $U = \min\{X,Y\}$

Let $X$ and $Y$ be independent random variables with common distribution function $F$ and density function $f$. Show that $V = \max\{X,Y\}$ has distribution function $\textbf{P}(V \leq x) = F(x)^{2}$ ...
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how to find pdf $f_X(x)$ from joint pdf $f_{X,Y}(x,y)$

enter image description hereI have joint PDF of $X$ and $Y$. $X$ and $Y$ are dependent random variable. I know that $X$ and $Y$ have the same distribution. it is hard to integrate joint PDF. are there ...
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Need help with transformation formula: how to get the pdf of $T$ given you know the pdf of $\ln(T)$

I know the answer is: $$f_T(t) = \frac{d}{dt} F_{\ln t}(\ln(t)) = f_{\ln t}(\ln t) \frac{1}{t}$$ which you can get by using cdfs and taking the derivative. $$F_T(t) = P(T \le t) = P(\ln(T) \le \ln(...
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1answer
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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}\...
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Let $M \sim \operatorname{Geometric}(p)$ and $ X \vert M = m \sim $ Uniform discrete on {1,m}

Let $M \sim \operatorname{Geometric}(p)$ and $ X \vert M = m \sim $ Uniform discrete on {1,m}. I want to find the density of X and then estimate the value of $p$ knowing one sample $X = x$. If ...
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1answer
25 views

Finding the law of X - Y when X and Y are not independent

I have the following joint density: $$ f_{X,Y}(x,y) =\frac{2}{(1+x)^3} (0<y<x) $$ I want to find the density of X - Y. Now, I found that $f_X(x) = \frac{1}{x} (0<x<\infty)$ and $ f_Y(y) = ...
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1answer
39 views

pdf of transformed variable

Given pdf $f_X(x)=\frac{x+2}{18}$ where $-2 < x < 4$, I wanted to find another r.v. $Y = \frac{12}{|X|}$. I think the support of $Y$ would be $3 < y < \infty$ but I wasn't super sure. I ...
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1answer
24 views

Finding characteristic function then density function is given

Random variable $\xi$ is distributed by symmetrical principle with density function $\frac{1}{2a} \mathcal{1}_{[-a,a]}(x),$ here $a>0$. I need to find characteristic function. I never seen ...
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$L^1$ between the sections of two functions

Let $f_1:\mathbb{R}^2 \to (0, \infty)$ and $f_2:\mathbb{R}^2 \to (0, \infty)$. Then consider the $L^1$ distance $$ \Vert f_1 -f_2 \Vert_1=\int_{\mathbb{R}^2}|f_1(x,y)-f_2(x,y)|dxdy. $$ Now, fix any ...
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Finding the probability density of the sum $Z=X+Y$ given a joint density function.

I'm given the joint density $$f_{X,Y}(x,y)= \left\{\begin{matrix}e^{-y}, \mbox{ } 0\leq x \leq y \le \infty \\ 0, \mbox{ otherwise}\end{matrix}\right.$$ and I'm told to the find the probability ...
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1answer
17 views

Marginal density for joint pdf of multiple random variables

I have the joint pdf $f(x_1,x_2,x_3)=12x_2 \;\mathrm f \mathrm o \mathrm r \; 0<x_3<x_2<x_1<1,$ and $0$ elsewhere. I want to find the marginal pdf $f_{x_3}(x_3)$. I know that to do this ...
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1answer
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Let $f(x,y) = \frac{1}{n!}(x-y)^n e^{-x} (0<y<x)$ be a joint density function

Let be $f(x,y) = \frac{1}{n!}(x-y)^n e^{-x} (0<y<x)$ a joint density function. I want to find the joint density function of $(U,V)$ where U = X and V = $e^{X-Y}$. Now, I already found that $X$ ~ ...
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Three-times differentiable density

Sorry for what might be an elementary question, but I can't find an intuitive explanation here or anywhere. I've been asked to consider a stochastic time series process $\{Y_t\}$ with three times ...
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show $F(a)=\sideset{_{-\infty}}{^a}\int f(x)dx$ is a distribution function

If $f$ is a function satisfying- $_{1)}$ $f(a)\ge 0$, $\forall a\in\mathbb{R}$ $_{2)}$ $\sideset{_{-\infty}}{^\infty}\int f(x)dx =1$ then, $f$ is a density function of the distribution function $...
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1answer
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Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique

The density function of the distribution function of a continuous random variable is not uniquely defined. A new density function can be obtained by changing the value of the function at finite ...
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Derive the probability density function of $Z = T_1 + T_2$.

Let $T_1$ be the waiting time until the first call in a call center and let $T_2$ be the waiting from the first call until the second call. Assume that both $T_1$ and $T_2$ have an exponential ...
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2answers
63 views

I can't figure out a specific step in the solution for the following problem: Let $X \sim U(-1,3)\,$ and $\,Y=X^4 $, find the PDF of Y

I am trying to study from pretty old class notes and encountered the following problem: $\text{Let } X \ \sim U(-1,3)\text{ and } Y=X^4 $, find the PDF of Y In the solution they used the following ...
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1answer
50 views

Given $ X \sim \exp(\lambda) $ find the density function of $Y=\frac{1}{1-X}$

Given $ X \sim \exp(\lambda) $ find the density function of $Y=\frac{1}{1-X}$ So I tried doing the following and gut stuck in the way: $ F_Y(y)=P(Y\leq y)=P(\frac{1}{1-x}\leq y)$ Now I though ...
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Probability density function calculation

In the image below, $h = \frac{2}{3}$. I have to calculate the Probability density function calculation for this image. I understand that there are diffrent values for diffrent $Xs$, for example I ...
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1answer
56 views

Computing marginal distribution

Assume $x$ is distributed with $F(x)=x^2$ for $0\leq x \leq 1$ and $c\mid x \sim U[0, \lambda \cdot x + 1-\lambda]$ for some $0 < \lambda < 1$. I am trying to find the marginal distribution of c....
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Density of a diffeomorphic image of a random vector

Let $X = (X_1, \dots, X_n)$ be a random vector and $f\colon \mathbb{R}^n\rightarrow \mathbb{R}^n$ an a.e.-differentiable mapping. If $f$ is a diffeomorphism, is it possible to derive the density of $...
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1answer
41 views

uniform distribution on [0,1] find function

Consider $X∼unif [0,1]$. Find a function $g: \mathbb{R} \longrightarrow \mathbb{R}$, such that g(X) has pdf $f(t) = \begin{cases} {t+1}, & \text{$-1 \leq t\leq 0$} \\ {1-t}, & \text{$0<t\...
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1answer
75 views

Finding $\mathbb{E}(\xi|\eta)$

Vector $(\xi, \eta)$ is evenly distributed in set $D=\{ (x,y): 0\leq x\leq 1, 0\leq y \leq 1, x+y \leq 1 \}$. I need to find $\mathbb{E}(\xi|\eta)$. So first of all I found the size of $D$. It is $...
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What claims and what Theorem we can prove using density property in Topology and functional analysis?

From wikipedia simple definition of dense set defined as ,In topology and related areas of mathematics, A subset A of a topological space X is called dense in X if every point x in X either belongs ...
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Inequality involving Monotone likelihood ratio and CDF ratio

This problem has really been bothering me and I have no idea whether the statement is true or not. So any help is appreciated. Suppose $f(x)/g(x)$ satisfies monotone likelihood ratio property and ...
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Prove that that PDF can be “factorized”, if its log satisfies a homogeneous PDE of the second order

The following problem is taken from "The Advanced Theory of Statistics, Kendall & Stuart, Volume 1, Second Edition, excercise 1.17: Let $f(x,y)$ be a probability density function. Prove that the ...
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1answer
40 views

The mutual density of $X,Y$ in $\{|t|+|s|<1\}$ is constant, are $X,Y$ independent?

Let $X,Y$ absolutely continuous random variables with density finctions $f_X,f_Y$. Assume that the mutual density $f_{X,Y}$ equals to a constant $c$ in $\{(t,s)\in\mathbb{R}^2:|t|+|s|<1\}$. Are $X,...
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28 views

Integrability of the quotient between two densities

Consider probability measures $(P_n)_{n=1}^\infty$ and $P$, all absolutely continuous with respect to the Lebesgue measure of $\mathbb{R}^d$ - say $\lambda$ - and all having the same support, $E \...
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What is the Probability density function of $X^2$ where X is an Uniform distribution

I'm a student and I'm studying random variables and very new to it. I was studying the Uniform distribution and in it, it calculates the Expected of $X^2$ by $$ E\left(X^2\right) = \int_{- \infty}^\...
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72 views

PDF of (X,Y) What is Z=X+2Y and joint probability density function of (Z,X)?

We have joint probability density function (PDF) of X and Y $f(x,y) = \begin{cases} e^{-x} & \text{for x > 0 ,y $\in$ (0,1), } \\ 0 & \text{in other case.} \end{cases}$ Let $Z = ...
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1answer
27 views

Why use indicator variables in p.d.f.s?

I am slightly confused about the use of indicator variables in probability density functions. For example, consider the density of $(X,Y)$ uniformly distributed on the unit disc. This density can ...
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1answer
24 views

Integral of KNN distribution

I encountered a question where I have to prove that the KNN density model does not define a proper distribution. Obviously I have to show that the integral of the function does not sum to $1$. The ...
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1answer
21 views

How do I determine the domain of marginal density functions?

I have the following density function. $$f(x,y)=\frac{1}{32}*(10-3x^2-y)\quad,-1<x<1; 0<y<2$$ When calculating the marginal density for $x$, $f_x$, I get $$\int_{0}^2\frac{1}{32}*(10-3x^2-...