Questions tagged [density-function]
Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.
1,712
questions
0
votes
3
answers
65
views
How to show that many functions (a hundred, a thousand) have the same shape an distribution of values over an interval?
I have functions that on iterval [0,1] all seem to look like this:
i.e. they have a zero around 0.4 +ve derivative from zero to 0.4 and around zero or slightly negative derivative up to 1.
I plan to ...
- 3
1
vote
1
answer
52
views
Does CDF must have value 0 at lowest possible input?
Suppose $F$ is the CDF of a real valued random variable. I know that $F(- \infty) = 0$, because the RV cannot take a value less than that.
But I was thinking of an RV whose value for sure comes from, ...
3
votes
2
answers
72
views
Exercise on finding probability density function
Let $Y_1$ and $Y_2$ by independent and uniformly distributed over the interval (0, 1). Find the probability density for $U = Y_1/Y_2$:
Solution:
$F_U(u) = P(U \le u) = P(Y_1/Y_2 \le u)$. Looking at ...
- 131
0
votes
0
answers
41
views
Distribution of a product of random variables
I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF
$$F_X(x) = \begin{cases}
F_X^1(x) & x \in (-\infty, a_1)\\
F_X^2(x) & x \in [a_1, a_2)\\
F_X^3(x) & x \...
- 1
2
votes
1
answer
122
views
Pushforward measure for Radon Nikodym equation
Consider the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and another probability measure $\mu$, on that same space, given by
$$\mu(A)=\int_A f(\omega) \mathbb{P}(d\omega)$$
Now let $X:\...
- 941
3
votes
2
answers
158
views
Estimating Smooth Density Field from Limited Sampled Data
I want to estimate a “density field”, specifically $P(y|x, m)$, for binary labels $y$ associated with 2D points characterized by spatial coordinates $m$ and additional spatio-temporal features $x$. ...
- 81
3
votes
1
answer
109
views
Uniform distribution over a triangle
Problem
Consider a triangle $T$ with vertices $V_1,V_2,V_3 \in \mathbb{R}^2$ and let
\begin{equation*}\begin{aligned}
y&=z+v\\
v&\sim\mathcal{N}(0, R)\\
z&\sim\mathcal{U}(T)
\end{aligned}\...
- 373
0
votes
0
answers
40
views
Uniform density over 2 segments [duplicate]
Background
Let $V_1, V_2 \in \mathbb{R}^2$ be the vertices of a segment and let $z$ be uniformly distributed over that segment. Now consider the random vector
\begin{equation*}
\begin{aligned}
y&=...
- 373
8
votes
2
answers
159
views
Sum of density functions
Consider four pdf $f_1(x), \ldots, f_4(x)$. For any $x$, $f_1(x) \neq \cdots \neq f_4(x)$.
Can we prove that $f_1(x) + f_2(x) \neq f_3(x) + f_4(x)$ for some $x$?
- 121
0
votes
0
answers
22
views
Determining Distribution for Conditional Probability
I have that the conditional probability density of $Y|X$ is as such
$f_{Y | X} \propto x^{y - 1}(1-x)^{n-y-1}\alpha^{n-y}\beta^{y}$
where $\alpha, \beta$ are constants in $(0, 1)$, $x$ is a random ...
- 13
0
votes
1
answer
31
views
Conditioning once or twice?
Let's say we have two random variables $Z \in \mathcal{Z}$ and $X \in \mathcal{X}$ with joint density $p_{Z,X}(z,x)$ with respect to a base measure. The density is assumed to factor as
$$ p_{Z,X}(z,x) ...
- 311
0
votes
1
answer
28
views
Changing bounds in marginal density
I have the function p(x,y) = 24x for 0<x, x+y<1, x<y.
I want to find the marginal density of Y, which means I have to integrate over x. My TA told me I have to split the area I want to ...
- 23
2
votes
0
answers
30
views
To what extent can likelihood methods be used for functional responses?
Let's suppose that we are working with a functional data set, $Y_i(t)$, $Y_i\in L^2[0,1]$, $1\le i\le n$. If we were working with univariate or even multivariate data set, likelihood methods would ...
- 1,373
2
votes
1
answer
57
views
Backtransforming a probabilistic forecast?
Let's say that we have a probabilistic forecast for the future percentage return of an asset in the form of a probability density, $\hat{R}_{t+1}$.
If our initial goal was to create a probabilistic ...
- 451
0
votes
0
answers
22
views
Finding the set for random variable transformations
I'm reading through the book "All of Statistics", and in section 2.12, regarding Transformations of Several Random Variables, the author lists three steps for finding the transformation. I ...
- 101