Questions tagged [density-function]
For questions on using, finding, or otherwise relating to probability density functions (PDFs)
0
votes
0answers
30 views
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 ...
0
votes
0answers
2 views
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 ...
1
vote
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
$\...
1
vote
0answers
43 views
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 &...
-2
votes
1answer
53 views
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 ...
0
votes
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!
...
0
votes
0answers
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:
$...
5
votes
1answer
21 views
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 >...
0
votes
0answers
26 views
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 ...
0
votes
2answers
39 views
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 ...
-1
votes
0answers
28 views
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
$$...
1
vote
2answers
52 views
Marginal distribution can't seem to work
Given $$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}
$$
The exercise itself doesn't ...
1
vote
0answers
36 views
Given the joint probability density function of the random variables $X$ and $Y$, determine $\textbf{P}(X < Y)$
The joint probability density function of the random variables $X$ and $Y$ is given by
$$f_{X,Y}(x,y) =
\begin{cases}
e^{-x-y} & \text{for}\,\,(x,y)\in(0,+\infty)^{2}\\
0 & \text{otherwise}
\...
1
vote
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
$$f_{X,Y}(x,y) =
\begin{cases}
\displaystyle\frac{1}{x^{2}y^{2}} & \text{if}\,\,x\geq 1\,\,\text{and}\,\,y\geq 1\\\\
\,...
2
votes
1answer
32 views
Naive question about continuous random variable: solution verification
The probability density function of the random variable $X$ is given by
$$f(x) =
\begin{cases}
a + bx^{2} & 0 < x < 1 \\
0 & \text{otherwise}
\end{cases}$$
If $\...
2
votes
2answers
29 views
Find PDF of Y from (constant) joint PDF
Given the following joint PDF:
$$
f_{X,Y}(t,s) =
\begin{cases}
\frac{2}{3} & 0\leq t\leq 1, -1\leq s\leq t \\
0 & otherwise
\end{cases}
$$
I need to find $f_y(s)$
So according to defintion:
...
0
votes
1answer
18 views
Find marginal density at a point [closed]
Given the following joint density function:
\begin{equation}
f (x,y) =
\begin{cases}
2& \text{} 0 \le x \le 1-y, 0 \le y \le 1\\
0 &\text{otherwise}
\end{cases}
\end{equation}
Find $f_{y}(...
1
vote
0answers
25 views
Given two independent identically distributed random variables, determine the distributions of $V = \max\{X,Y\}$ and $U = \min\{X,Y\}$
Let $X$ and $Y$ be independent random variables with common distribution function $F$ and density function $f$. Show that $V = \max\{X,Y\}$ has distribution function $\textbf{P}(V \leq x) = F(x)^{2}$ ...
0
votes
0answers
43 views
how to find pdf $f_X(x)$ from joint pdf $f_{X,Y}(x,y)$
enter image description hereI have joint PDF of $X$ and $Y$. $X$ and $Y$ are dependent random variable. I know that $X$ and $Y$ have the same distribution. it is hard to integrate joint PDF. are there ...
1
vote
0answers
20 views
Need help with transformation formula: how to get the pdf of $T$ given you know the pdf of $\ln(T)$
I know the answer is:
$$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(...
0
votes
1answer
22 views
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}\...
0
votes
0answers
30 views
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 ...
1
vote
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) = ...
1
vote
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 ...
0
votes
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 ...
2
votes
0answers
57 views
$L^1$ 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 ...
-1
votes
1answer
52 views
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 ...
1
vote
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 ...
1
vote
1answer
22 views
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 $X$ ~ ...
0
votes
0answers
14 views
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 ...
1
vote
0answers
27 views
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 $...
0
votes
1answer
20 views
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 ...
0
votes
0answers
29 views
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
21 views
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 ...
0
votes
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....
0
votes
0answers
13 views
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 $...
0
votes
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\...
0
votes
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 $...
0
votes
0answers
50 views
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 ...
2
votes
1answer
19 views
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 ...
0
votes
0answers
25 views
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 ...
0
votes
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,...
0
votes
0answers
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 \...
1
vote
0answers
35 views
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 $$ E\left(X^2\right) = \int_{- \infty}^\...
1
vote
0answers
72 views
PDF of (X,Y) What is Z=X+2Y and joint probability density function of (Z,X)?
We have joint probability density function (PDF) of X and Y
$f(x,y) =
\begin{cases}
e^{-x} & \text{for x > 0 ,y $\in$ (0,1), } \\
0 & \text{in other case.}
\end{cases}$
Let $Z = ...
0
votes
1answer
27 views
Why use indicator variables in p.d.f.s?
I am slightly confused about the use of indicator variables in probability density functions.
For example, consider the density of $(X,Y)$ uniformly distributed on the unit disc. This density can ...
1
vote
1answer
24 views
Integral of KNN distribution
I encountered a question where I have to prove that the KNN density model does not define a proper distribution. Obviously I have to show that the integral of the function does not sum to $1$.
The ...
0
votes
1answer
21 views
How do I determine the domain of marginal density functions?
I have the following density function.
$$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-...
