Questions tagged [density-function]
For questions on using, finding, or otherwise relating to probability density functions (PDFs)
1,346
questions
0
votes
1answer
20 views
Transforming random vector - checking correctness of solution
The task is to calculate two-dimentional probability density of $[X,Y]$ if we know that:
$$
R \sim U(0,1), \quad \Phi \sim U(0, 2\pi),
$$
$R$ and $\Phi$ are independent and
$$
X := R \cos(\Phi), \...
-1
votes
1answer
13 views
PDF for a chi-squared distribution divided by its degrees of freedom
What would the pdf be for a chi-squared distribution divided by its degrees of freedom. Would it be the normal pdf / degrees of freedom, or is it more complicated?
-4
votes
1answer
44 views
Compute $f_X(x)$ and $f_Y(y)$. And are X,Y independent?
Let $(X,Y)$ be a random vector with joint pdf $f_{X,Y}(x, y)$ = $1_{[−1/2,1/2]^2}(x, y)$. Compute $f_X(x)$ and $f_Y(y)$. And are X,Y independent?
I got:
$f_X(x)$ = $\int_{-0.5}^{0.5} dy$ = $1$
$...
0
votes
0answers
20 views
Continues joint distribution marginal PDF calculation
I have a joint function f(x,y) and i have two regions as stated on the above graph but i don't know if i am going to calculate Marginal PDF of X and Y differently on both regions and because ...
0
votes
1answer
21 views
$f, g$ are probability density functions of an normal distribution N(0,1), prove h is $N(0,\sqrt 2)$
I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function.
When $f$ and $g$ are $exp(\lambda)$ it's solved by $\int_{0}^{...
0
votes
1answer
29 views
How to define a PDF for data with unkown distribution?
I have a dataset containing real values and I want to define the PDF associated. Is there any method to find out the PDF for data with unknown distribution?
0
votes
1answer
28 views
Help understanding approximation of integral of pdf
Assume that $f$ is the pdf of a continuous random variable $X:\Omega\to\mathbb{R}$. Let $\varepsilon>0$. Then:
\begin{equation*}
\mathbb{P}\left(X\in\left[x-\frac{\varepsilon}{2},x+\frac{\...
0
votes
0answers
14 views
density function of $W(2/3)$
$W(t)$ - Wiener process on $(\Omega, F, P)$ and measure $Q$ such that $dQ=e^{W(1)-\frac{1}{2}}dP$. How to find density fuction of $W(2/3)$ with respect do measure $Q$?
0
votes
1answer
24 views
$f, g$ are probability density functions of an exponential distribution, prove h is $\gamma (\lambda ,2)$
I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function.
Then is asked: $f, g$ are probability density functions of an ...
0
votes
1answer
37 views
PDF of $Q$ Random Variable
Let $X\sim N(0,25)$, $Y\sim N(10,100)$, $Z\sim N(-10,50)$ and $Q=\tan^{-1}\left(\frac{Z}{\sqrt {X^2+Y^2}}\right)$
When I simulate $Q$ random variable with Monte Carlo method, I'm getting this ...
0
votes
1answer
37 views
How to find the PDF of a function of two random variables
Y is a uniform continuous random variable between [0,L] and X is a uniform continuous random variable given Y=y between [0,y]. What is the PDF of Z = X/Y.
1
vote
2answers
32 views
Finding a density function of the sum of two random using the convolution
The following problem is from the book "Probability and Statistics" which is part of the Schaum's outline series. It can be found on page 71 and is problem number 2.74. It is under the section ...
0
votes
1answer
31 views
Find the density of the median of $2n$ i.i.d $U([0,1])$ random variables
As a part of a probability problem I found the PDF and CDF of the $i^{th}$ order statistic in a sample.
When told that $$X_1,...,X_n{\sim}^{i.i.d}F$$ where F is countinuous, so I got to the conclusion ...
0
votes
1answer
20 views
Covariance of a joint PDF with a min function.
I have to find the covariance of the following PDF
$$f(x,y)=\begin{cases} 3\min\{x,y\}&, \text{ if }\ 0<x,y<1 \\ 0 &, \text{ otherwise} \end{cases}$$
Therefore I need to find $E(X)$ ...
0
votes
0answers
21 views
OOD detection using softmax
So I was working with Hendrycks Baseline method (https://arxiv.org/pdf/1610.02136.pdf) and I'm completly lost over this topic.
I understand why this is working, we just make a normalisation utilizing ...
0
votes
1answer
25 views
PDF of function of random variable in multidimensional case proof
I read the wiki page on PDF and I got stuck at the proof of relation between pdf $g$ of a function $\textbf{y}$ and pdf $f$ of its random variables $\textbf{x}$:
$$g({\bf{y}}) = f({H^{ - 1}}({\bf{y}}...
0
votes
0answers
20 views
How to find the marginal PDF's of Bivariate Gamma Distribution
Let $X$ and $Y$ be two random variables with joint PDF
$$f_{XY}(x,y)=\frac{1}{\Gamma(a)\Gamma(b)}x^{a-1}(y-x)^{b-1}e^{-y},\quad0<x<y,\quad\text{Bivariate Gamma Distribution}$$
where $a$ and $b$ ...
0
votes
1answer
37 views
Finding distribution of $W=\frac{X}{Y-X}$
$X$ and $Y$ are two random variables, and
$f_{X,Y}(x,y)=6(y-x)$, $0 < x < y < 1$
If $W=\frac{X}{Y-X}$, how can I find $f_W(w)$ and show that $W$ and $Y$ are independent? I've calculated
$A=...
1
vote
1answer
30 views
How to calculate the joint probability: $\Pr \left( \tfrac{g_1}{g_3} \geq \theta_1, \tfrac{g_2}{g_3} \geq \theta_2, g_3 > \theta_3 \right)$?
Question:
How to calculate the following?
$$\Pr \left( \dfrac{g_1}{g_3} \geq \theta_1, \dfrac{g_2}{g_3} \geq \theta_2, g_3 > \theta_3 \right),$$
where $g_i, i \in \{1, 2, 3\}$ is an exponentially ...
0
votes
1answer
20 views
Marginal distributions for a standard bivariate Cauchy distribution
Consider the following:
Let $X$, $Y$ be two jointly continuous random variables with joint PDF
$$f\left(x,y\right)=\frac{c}{2\pi}\frac{1}{\left(c^{2}+x^{2}+y^{2}\right)^{\frac{3}{2}}}\quad\text{(...
2
votes
1answer
41 views
PDF with a finite second moment converges to zero?
Suppose for a probability density function $p(x)$, $x\in \mathbb{R}$, we have $\int_{-\infty}^{+\infty} x^2p(x)dx < \infty$. Can we claim $\lim_{x\rightarrow\infty} p(x) = 0$?
I think it boils ...
0
votes
1answer
21 views
Joint density function of $X$,$Y$,$Z$, independent random variables with exponential distribution
Let $X$,$Y$,$Z$ be independent random variables with exponential distribution of parameter $\lambda$, then $X,Y,Z$ ~ $\xi(\lambda)$.
Is it true that $f_{X, Y, Z}(x,y,z)=\lambda^{3}e^{-\lambda (x+y+z)}...
0
votes
1answer
18 views
If $X$ is symmetric at $a$, then $Y= X + a$ is symmetric at $0$
Suppose $X$ has a density function $f$ that is symmetric about $a$. Let $Y = X + a$. Show that the density function $g$ of $Y$ is symmetric about $0$.
Setting $f(x) = g(x-a)$ gives you the result ...
2
votes
1answer
42 views
When and why do formulae involving sums over $x_i$ change to formulae involving $X$ in statistics? Specifically when dealing with likelihoods.
I've been reading up on stats recently and a question I'm working through involves calculating the log-likelihood of a distribution w.r.t a parameter $\beta$.
From my understanding, for some ...
1
vote
0answers
33 views
Find the joint and marginal distributions of $X + Y$ and $Y - X$. [closed]
Let $f_{X,Y}(x,y)=k(x+y) I_{(0,1)}(x) I_{(0,1)}(y) I_{(0,1)}(x+y) $
Find the joint and marginal distributions of $X + Y$ and $Y - X$.
so $U=X + Y$ and $V=Y - X$
doing the Jacobian
$J=\frac{1}{2}...
1
vote
1answer
48 views
Radon-Nikodym Derivative of a Mixed Distribution
When we have a continuous distribution $F_X(x)$, we can take the Radon-Nikodym derivative (RND) of the probability measure with respect to Lebesgue measure to get the density $f_X(x)$.
When we have ...
1
vote
0answers
21 views
Find $f_{X|Y=y}$ and $f_{Y|X=x}$ when $f_{X,Y}(x,y)=C\cdot \text{exp}\{-(1+x^{2})(1+y^{2})\}$
Consider the density
$f_{X,Y}(x,y)=C\cdot e^{-(1+x^{2})(1+y^{2})},\quad -\infty <x,y<\infty$
$C$ is a normalizing constant.
What are the conditional densities $f_{X|Y=y}(x)$ and $f_{Y|X=x}(y)$...
0
votes
3answers
51 views
If $X$ is a random variable, and $Y= 2X$, then why isn't it enough to multiply the density function of $X$ by $2$ to find the density function of $Y$?
This may be a dumb question, and I've tried searching online for answers, but I can't seem to wrap my head around it.
So say I have a random variable $X$ and $Y = 2X$. Now I want to find the density ...
0
votes
1answer
17 views
Showing a minimal sufficient statistic
If we have common density $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$
with $x\in(0,1)$, $\theta>0$ and $\textbf{X}=(X_1,...,X_n)$ is a random sample.
Then how can we show that the ...
1
vote
0answers
14 views
The Fractional Derivative and Stochastic Processes/ Fokker Planck equation
In this question I am interested in Fractional Derivatives and the Fractional Fokker Planck Equation. Note I only mention fractional in time, however if someone wants to reference to fractional in ...
0
votes
0answers
8 views
Quantile functions of absolutely continuous distributions
Let $F:\mathbb{R}\to \mathbb{R}$ be a cumulative distribution function. its quantile function is defined by
\begin{equation}
q(t):=\inf \{x \in \mathbb{R} \ | \ F(x) \geq t \}
\end{equation}
A ...
0
votes
0answers
28 views
sum of folded normal distribution
I have $n$ iid normal random variables $X_i\sim N(0,1)$, and their folded versions $Y_i=|X_i|$. Now I need the cdf, or the density of the sum
$$
Y_1+Y_2+\dots+Y_n
$$
or at least a good approximation (...
1
vote
1answer
47 views
What is the density of Z?
if $f_{x,y}(x,y)= I_{(0,1)}(x) I_{(0,1)}(y) $ find the density of $Z$, where
$$Z= (X+Y) I_{(-\infty,1]} \hspace{.2cm} (X+Y) + (X+Y-1)I_{(1,\infty)} \hspace{.1cm} (X+Y) .$$
I know that $V=X+Y$ has ...
0
votes
1answer
28 views
Polar coordinates and Jacobian
Let $(V,W)$ a point in the circle of unity radius chosen in accordance with the following rules. First, let $R$ a random number uniform in $(0,1)$. Second, you choose a point $X$ on the circumference ...
0
votes
1answer
22 views
0
votes
1answer
55 views
Conditional and joint distribution of the sum of exponential RVs
Let $X_1,X_2,...,X_n$ be i.i.d. $Exp(\lambda)$ random variables and $Y_k =\sum^{k}_{i=1}X_i$, $k = 1,2,...,n$.
a) Find the joint PDF of $Y_1,...,Y_n$.
b) Find the conditional PDF of $Y_k$ ...
0
votes
0answers
8 views
Density function from chart
I was trying to solve a problem for my class, and I don't quite get how should I approach it. I'm supposed to find the distribution function and the moment generating function for a density function ...
0
votes
2answers
39 views
Is this a valid pdf? Continuous r.v. with “weird” pdf
The pdf of a continuous random variable is given by:
$$ f_X(x) = 1/3,~ 0 < x < 1 $$
$$ f_X(x) = 2/3,~ 1 < x < 2 $$
$$ f_X(x) = 0,~ \text{otherwise} $$
My problem with this continuous ...
0
votes
1answer
17 views
Density function and constant k
I was trying to solve a problem, but I got stuck. Suppose that a given demand function follows that P = 100 - Q (where P is price and Q is quantity), and the problem says that the variable Q has a ...
0
votes
0answers
81 views
How can I show $ \int_{-\infty}^\infty\int_{-\infty}^{\infty} f(x,y)\, dx\,dy = 1 $?
I'm stuck on this one..
$$ \displaystyle\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \pi e^\frac{-(x^2+y^2)}{2} dxdy = 1 $$ knowing $f(x,y) > 0$ ?
I mean i can try $$ \pi \displaystyle\int_{-\...
1
vote
2answers
52 views
Is $g(X|Y=y)$ equivalent to $g(X)|Y=y$?
Let $X$ and $Y$ be random variables with marginal pdfs $f_X(x)$ and $f_Y(y)$, respectively, and joint pdf $f_{X,Y}(x,y)$. Then for all $y$ such that $f_Y(y) \neq 0$ define the function
$$f_{X|Y=y}(x) =...
0
votes
1answer
101 views
Given a multi-dimensional sample, how do I build a distribution density coefficient? [closed]
Given a sample $X=\{\vec{x}_1, \dots,\vec{x}_l\}$ where $\vec{x}_i \in \mathbb{R}^d$ with $d>3$, I would like to know if it's possible to have and index that is inversely proportional to the ...
2
votes
2answers
35 views
How to derive the probability density function (PDF) of a continuous random variable from a set of data?
I am interested to derive an expression for the probability density function (PDF) of a continuous random variable from a given set of data. To further explain, let us consider that we have the data ...
0
votes
2answers
60 views
Probability density of analytical function on 3 random variables
I know some methods to obtain the probability distribution of functions on a random variable:
CDF method: If $X$ is a random variable and $Y=f(X)$, then computing the cumulative distribution ...
1
vote
1answer
21 views
General condition for PDF of a random variable, so that it is self-inverse
I am supposed to find a general condition for a PDF of a random variable X, so that the distributions of X and 1/X are the same. I showed this for standard Cauchy distribution using the formula
$ g(y)...
0
votes
1answer
32 views
How do I handle this probability density function with a Jacobian?
"Suppose X and Y are independent random variables, each exponentially distributed with parameter $\lambda$. Determine the probability density function for $Z=\frac{X}{Y}$."
Here is what I have so far:...
1
vote
1answer
78 views
Finding pdfs of $\frac1{X^2}$ and $\frac{1}{4}\left(\frac1{X^2}+\frac1{W^2}\right)$ where $X,W$ are independent $N(0,1)$
$X,W$ are independent random variables, both $N(0,1)$, i.e. $f_X(x)=f_W(x)= \frac{1}{{\sqrt{2\pi}}}e^{-\frac{x^2}{2}}$.
Find PDF of $Y:=\frac{1}{X^2}$
Find PDF of $\frac{1}{4}\left(\frac{1}{X^2}+\...
0
votes
0answers
51 views
How to inverse the laplace transform $\frac{1}{\cosh(5\sqrt{s})}$?
Let $X$ be a random variable with $ E[e^{-sX}]=$ $\frac{1}{\cosh(5\sqrt{s})} $ and density function $f$. How to give a formula for $f$?
2
votes
1answer
56 views
PDF of $Y=X(X-1)$ when $X$ has a piecewise PDF
I have to solve the following problem:
Let $X$ be a continuous random variable with PDF
$$f(x)=
\begin{cases}
x+1, &-1\leq x<0\\
1-x, &0\leq x\leq1\\
0, &\text{otherwise}
\end{cases}.$$...
-1
votes
2answers
29 views
Can a probability density funtion have a negative domain?
I've only ever seen domains stretching from $X\ge0$.
I have a question where:
$F(x)=cx^2$, Domain: $− 1 ≤ x ≤ 1$
When finding value of $c$, would I only need to integrate $0$ to $1$. And does that ...
