All Questions
Tagged with density-function joint-distribution
87 questions
4
votes
1
answer
543
views
An impossible distribution
Some days ago another user posted a question which was something like this:
$$ A \sim U(0,4)$$
$$B \sim U(0,6)$$
$$A - B \sim U(-4,4)$$
The question was originally to find the distribution of A ...
- 304
0
votes
0
answers
48
views
Converting an integral into a probability of some event
Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$
$$$$Consider the ...
- 217
3
votes
1
answer
177
views
Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian
Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$.
I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
- 33
0
votes
0
answers
39
views
Probability of a vector. Is my notation correct?
Question 1:
Let us first consider the univariate case:
Suppose we have $Y\in\{0,1\}$, suggesting that $Y$ is Bernoulli variable (and hence discrete) and $X\in \mathbb{R}$, then we know that
\begin{...
- 1,226
3
votes
3
answers
337
views
Distribution change of variables
I'm working on distributions for a physics problem and I am quite stumped on how to proceed properly.
The problem is as follows: I start at the point $(0,0)$ and go a distance $\eta$ in x direction $(\...
- 95
1
vote
0
answers
119
views
What is the probability mass function of Rock, Paper, Scissors?
I was curious about the statistics behind the game of rock, paper, scissors. Let's say n people are playing, where n is greater than or equal to 2. If when all n people reveal their play and only 1 ...
- 793
0
votes
1
answer
51
views
Finding probability of all success for in an order statistic
𝑓(𝑦) = 5𝑦^4; 0 ≤ 𝑦 ≤ 1
A group of 3 friends order small cups of soda, from the soda dispenser. If
the 3 small cups are considered a random sample from the dispenser fills, find the probability
...
3
votes
0
answers
84
views
Density function of a dependent sum of products of normal random variables
Say we have a random variable
$$
X = A_0 A_1 + A_0 A_2 + A_1 A_2,
$$
which consists of normally distributed independent random variables $A_0, A_1, A_2 \sim \mathcal{N}(0,1)$ with probability ...
- 31
1
vote
1
answer
191
views
How to get the pdf of a joint distribution from its kernel?
How could you obtain the pdf of a joint distribution from a multivariate kernel?
- 89
6
votes
2
answers
2k
views
How to write a joint kernel density of two random variables with known individual densities?
Consider two random variables $X$ and $Y$ with densities
$${f}_1(x) = \frac{1}{n_1h_1} \sum\limits_{i=1}^{n_1}K\left(\frac{x-u_i}{h_1}\right) ~~~~\text{and} ~~~~ {f}_2(y) = \frac{1}{n_2h_2} \sum\...
- 765
0
votes
1
answer
218
views
How do I find the constant of a continuous joint probability distribution function in R?
I'm wondering how to set up the calculation for a double integral to solve for the value of c for the problem below.
Consider the joint probability density function $f_{XY} (x, y) = c(x+y)$ over the ...
1
vote
1
answer
42
views
Probabilities and joint pdf [closed]
I'm supposed to compute $P(Y>1/2)$ for the joint pdf
$$f_{X,Y}(x,y)=\begin{cases}12(y-x^2),&0<x<y<1\\0,&\text{elsewhere.}\end{cases}$$
and figured that I can do this if I use the ...
user331219
0
votes
0
answers
79
views
Can we compute the joint distribution given the marginal distribution (and other informations)?
I have only studied the very basics of statistics, but I always wondered ever since I first studied the joint distribution of two random variables if it's possible to get the joint distribution from ...
- 101
0
votes
0
answers
25
views
Expected value of X given Y is less than some constant [duplicate]
Here is the problem I'm trying to work out: Let $v_b,v_s$ be jointly normally distributed random variables with pdf $f(v_b,v_s)$. I want to work out $E[v_b|v_s\leq\pi]$ for some constant $\pi$. Here ...
- 1
0
votes
0
answers
45
views
If A and C are independent random variables, calculating the pdf of AC using two different methods [duplicate]
Suppose $A$ and $C$ are uniformly distributed over $(0,1)$ and are independent random variables. Then, I found the pdf of $AC$ using this method:
$$f_{AC} (a,c) = f_A (a) f_C(c) = \begin{cases}
...
- 223
0
votes
0
answers
227
views
joint PDF of continuous and discrete random variables
Given exponential a random distribution X with PDF $f_X(x)=\lambda e^{-\lambda x}$ and a random variable Z with the PMF $p_Z[z]=0.5, z= \pm1$, I am trying to find the PDF of $Y=ZX$ (I also know that Z ...
- 1
1
vote
1
answer
1k
views
Distribution of Functions of One or Two Random Variables
I just wanted to confirm my understanding related to the distributions of functions of random variables. Can someone please tell me if all of my points are correct and make sense? I also have an ...
- 175
2
votes
1
answer
2k
views
Expected Value for 2 Random Variables with Joint Probability Distribution
I have trouble with determining the domain for integration in the case of having a joint pdf when one variable depends on the other. There are two examples I don't quite understand, and hopefully, you ...
- 175
1
vote
0
answers
101
views
If $Y$ is continuous and $X$ is discrete, how to write the joint density of $(Y,X)$?
If $Y$ is continuous and $X$ is discrete with a finite number of points in the support, how to write the joint density of $(Y,X)$?
For example, to write the joint density function evaluated at $(Y,X)=(...
- 3,050
2
votes
1
answer
70
views
Domain problem when calculating marginal density
I have the following homework assignment: the life expectancy $X$ of a lamp has exponential distribution with rate $\lambda$. The rate depends on the production proccess, such that its population can ...
0
votes
0
answers
210
views
Computation of the density of the ratio of two random variables
Background:
For two continuous random variables, $X$ and $Y$, the density of $Z := \frac{X}{Y}$ is given by
\begin{equation}
p_Z(z) = \int_{-\infty}^\infty \lvert y\rvert\, p_{XY}(zy, y) \, \text{...
- 111
0
votes
1
answer
106
views
Facing difficulties in Cauchy PDF problem from harvard stats 110 book [duplicate]
I have two doubts:
If X and Y are independent then PDF of X/Y is simply PDF of X(by independence) I don't know where am I interpreting wrong, Please correct me.
How X/Y and X/|Y| are identically ...
0
votes
1
answer
145
views
Do copula describe multivariate distributions more accurately than their moments?
Moments approach:
A common way to characterize and describe the density (pdf) of a random variable is to only look at the mean and standard deviation ($\mu_1$, $\sigma_1$) of its pdf. This mentality ...
- 4,049
4
votes
2
answers
373
views
What is the copula of a variable with itself?
In Sklar's theorem for joint probability functions,
$$f(x,y) = c(F_X(x), F_Y(y)) \cdot f(x) f(y)$$
the copula is $c(\cdot)$ of variables $X$ and $Y$, while $f(\cdot)$ are their marginal distributions.
...
- 4,049
0
votes
0
answers
184
views
Simple function of two random variables, $Z = \frac{Y}{X}$ where Y, X~U(0,1) and independent [duplicate]
I must have missed something important in the formulation of the problem. Can you please help clarify how should the following simple problem be formulated and where my mistake is.
Let $Z = \frac{Y}{X}...
- 215
1
vote
1
answer
681
views
Mixed Joint Probability
In the wikipedia definition they give the example of a logistic regression problem to predict $P(Y=y\,|\,X=x)$ where $Y$ is binary (discrete) and $X$ is a continuous random variable. Then the mixed ...
- 113
5
votes
1
answer
889
views
Why copula based on CDF instead of PDF
I do understand the mathematic behind probability density function( PDF) and cumulative distribution function (CDF). My problem starts when I try to understand why copula relies on CDF and not on PDF. ...
- 1,680
1
vote
1
answer
291
views
Conditional probability density from probabilities
I am trying to understand conditional probablility densities in relation to the conditional probablilities. From the Measure-theoretic definition on Wikipedia, if $X$ and $Y$ are non-degenerate and ...
- 13
1
vote
1
answer
46
views
Marginal distribution
A loss distribution has PDF - $f(x) = 1/x^2$, for $x > 1$
An insurer finds that the time in hours it takes to process a loss amount x has a uniform distribution on the interval $(\sqrt x, 2\sqrt x)$...
3
votes
1
answer
102
views
Finding P(a< u(X,Y) <b) given a rectangular support
>The continuous variables X and Y have the following joint pdf $f(x,y) = x + y, 0<x,y<1.$
Determine $P(0.5<X+Y<1.5)$.
I know that the support of x and y is rectangular, hence they are ...
- 33
5
votes
2
answers
2k
views
If $X=\sin\Theta$ and $Y=\cos\Theta$ with $\Theta$ uniformly distributed, how can I compute the joint pdf of $(X,Y)$?
I have a random variable $\Theta$ uniformly distributed between $[-\pi ,\pi]$, two functions $X=\sin\Theta$ and $Y=\cos\Theta$. I know that $X$ and $Y$ are uncorrelated but not independent. I want to ...
- 83
1
vote
0
answers
14
views
bivariate conditional joint sensor model
I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables.
...
- 11
4
votes
2
answers
7k
views
Determining Independence of two random variables from joint density function
I know that to determine whether $X$ and $Y$ are independent I have to find the marginal distributions of $X$ and $Y$. I've already found the marginal probability density function $f_Y$, but I'm ...
- 43
3
votes
1
answer
353
views
Finding the joint distribution of a subset of random variables from the joint distribution of the superset?
If I have a set of random variables $X_1\dots X_n$ with joint density $f(x_1,\dots,x_n)$, if I wanted the joint density of any (say) two random variables $X_i$ and $X_j$, can I find this using the ...
- 1,209
1
vote
1
answer
2k
views
Normalizing the joint probability density
I computed the kernel estimators for the copula density for two random variables using:
library(kdecopula)
kde.fit <- kdecop(u)
As the values of density can be greater than one I was wondering ...
- 97
1
vote
1
answer
742
views
Find joint pdf table of two discrete independent random variables $X$ and $Y$
Given the pdfs of two discrete independent variables $X$ and $Y$, write the joint pdf. There is a property that $ if\ \ p_{XY}(x,y) = p_X(x)p_Y(y) \ \forall i,j
\Rightarrow \text{X,Y are ...
3
votes
2
answers
535
views
Joint distribution of random vector and a linear combination of it
If $\mathbf{X} \sim \mathcal{N}_n(\mathbf{\mu}, \mathbf{\Sigma})$, what is the joint distribution of $(\mathbf{X}, \sum_{i=1}^n c_i X_i)$ where $c_i$ are constants?
I've tried to work this out, but ...
- 305
2
votes
2
answers
178
views
Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?
What the title says!
My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
- 61
0
votes
1
answer
98
views
Calculation of joint PDF
we have the joint PDF of two RVs $X$ and $Y.$
we also have two RVs $U = f(X,Y)$ and $V = g(X,Y),$ where
$f$ and $g$ are two variable functions.
How can I calculate the joint PDF of $U$ and $V$?
for ...
1
vote
1
answer
868
views
Example: Writing the joint PDF $f(x, y)$ as the product of a marginal and a conditional probability function
I am presented with the following notes on Bivariate distribtions:
If we can write the joint probability density function $f(x, y)$ of a pair of random variables $(X, Y)$ as the product of a marginal ...
- 2,204
0
votes
1
answer
32
views
Bivariate Dist Study Question Help - determine joint PMF and P( ... )
I am in a prob. models class. Current module is on Bivariate and Multivariate Distributions. The question below has me stumped though. It is from a study guide and I would like to know the answer ...
- 21
1
vote
0
answers
41
views
How to find pdf of transformed r.v using joint distribution
I'm intrigued by the following idea but I don't know how to do it.
If I have a r.v. $x$ with given distribution $f_X(x)$ and I have a second variable $y=2x$. The goal is to find $f_Y(y)$.
I know the ...
- 255
5
votes
2
answers
254
views
Intuitive explanation of "density generators"?
I was reading through Meucci's Risk and Asset Allocation (2005), when I happened upon the concept of a "density generator", which I have not been able to find good explanations for anywhere online, ...
- 662
1
vote
1
answer
498
views
Computing a marginal distribution of a joint involving a delta function
Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral:
$$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$
There are a few conditions:
$...
- 286
5
votes
2
answers
889
views
Find mgf from joint pmf
The joint pmf of random variables $ X$ and $ Y$ is given by
$$p_{XY}(x,y)=
\begin{align}
& \frac{e^{-2}}{x! (y-x)!}\quad\text{if}\,\,\,x= 0,1,...y,\ y=0,1,... \\
\end{align}
$$
Find its mgf.
\...
- 1,397
0
votes
1
answer
38
views
Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]
Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x,
0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$,
...
- 115
3
votes
1
answer
9k
views
Finding the joint CDF using the joint PDF; why can't I do this?
Find the joint CDF of the independent random variables $X$ and $Y$, where
$f_X(x)=x/2, 0\le x \le 2, $ and
$f_Y(y)=2y, 0 \le y \le 1$.
To do this, we can find the CDF separately for each of the ...
- 115
0
votes
3
answers
2k
views
Compute $P(Y<3X)$ using joint PDF
I'm given a joint pdf
$f_{X,Y}(x,y)=2e^{-x-y}, 0<x<y, 0<y $
and asked to compute $P(Y<3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this ...
- 115
1
vote
1
answer
450
views
Finding the CDF given marginal PDF's; setting bounds
In this question, I'm having a hard time understanding how specifically to set the bounds for the CDF.
Let $X$ and $Y$ be independent variables. Find the CDF of $W=Y/X$ using
the marginal PDFs
...
- 11
1
vote
1
answer
337
views
PMF and independence with two discrete random variables?
Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
- 331