Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,024 questions
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Bounds for number of coin toss switches
I toss $n$ biased coins and I want to count the number of times you get a H followed by a T or a T followed by a H. I call these switches. So for example if I get HHTTHTHHHT then I have $5$ switches ...
- 11
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2
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305
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Properties of the Euler Discretization of a diffusion
Let $X$ be a continuous 1-d diffusion:
$$
dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x.
$$
W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties.
Let $Z^n_1,Z^...
- 943
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1k
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Probability of an edge appearing in a spanning tree
Hi guys, let's say I have a connected, undirected graph with many nodes. I am interested in finding the probability that an edge appears in any spanning tree of the graph. I could apply some of the ...
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329
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Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
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155
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Again a question related to uncorrelatedness and independence.
Consider a random vector $\mathbf{Z}$ of length $N$ whose elements are i.i.d. and zero-mean. We construct two new scalar random variable $X$ and $Y$ from $\mathbf{Z}$ as follows:
$X = \langle\mathbf{...
- 197
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1
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426
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Lower bounds for partial sums of multiplicative functions
The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
$$F(s)=\...
- 2,480
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1
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197
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Distribution wanted
I have a centered random variables $Y_1,Y_2$ which first 10 moments are given respectively by
$$ 0, 1, 0, 6, 0, 90, 0, 2520, 0, 113400 $$
$$0, 1, 0, 32/3, 0, 36847/100, 0, 436879364/15435, 0, ...
- 1,491
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2
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630
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Units in Ornstein-Uhlenbeck(OU) process
Take an OU process characterized by
X(0) = x
dX(t) = - a X(t) dt + b dW(t)
where a,b >0. The parameter a is usually interpreted a dissipative term, and b is a ...
- 249
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2
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126
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Unique coupling
Let $X$ be a Polish metric space, and let $\mu,\nu$ be two Borel probability measures on $X$, when is the product measure the only coupling of $\mu$ and $\nu$. More formally, let $$\Gamma(\mu,\nu):=\{\...
- 245
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337
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How far does a random walker travel before returning to the origin?
Consider a (continuous time) simple symmetric random walk on $\mathbb Z$, starting from the origin. Let us denote this random walk by $\{X_t: t\geq 0\} $. It is well known that this random walk is ...
- 59
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230
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Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
Let
$X := \mathbb R^n$,
$C_b(X)$ the space of all real-valued bounded continuous,
$C_c(X)$ the space of all real-valued continuous functions with compact supports, and
$C_c^\infty(X)$ the space of ...
- 657
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59
views
Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$
A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
(1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
(2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
(3) $\phi$ is ...
- 6,853
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109
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Do these $L^p$ type statistics characterize distributions?
Define for real valued random variable $X\in L^p$, the $p$-statistic
$$X_p:=\arg\min_{c\in \mathbb R}E[|X-c|^p].$$
For example $X_1$ is the median of $X$, $X_2$ is the mean of $X$ and also $X_\infty$ ...
- 1,904
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370
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Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$
Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse.
It is evident that
$$
\mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p,
$$
where $X\sim N(0,1)$.
Is ...
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2
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Mathematics Roadmap [closed]
I immediately apologize for my English, Google translator is my assistant. I couldn't find the information in my own language.
My question is addressed to people who understand mathematics. I hope for ...
- 25
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243
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Integral form of expectation with respect to complex random variables [closed]
Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
- 13
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534
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Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
- 6,853
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1
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196
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Sufficient conditions for finite mean of a non-negative random variable
Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition:
$$\lim_{x\rightarrow\infty} x(1-F(x)...
- 79
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313
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Some doubts on proof of pathwise uniqueness of a stochastic differential equation
I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...
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519
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Bounds on variance of sum of dependent random variables
Let $x_1, \ldots, x_n$ be possibly dependent random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. ...
- 153
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109
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Can we say that $f_\infty(t)=g_\infty(t) \text{ a.e}$?
Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ and $\{g_n\}$ two uniformly integrable sequences such that:
$$
\qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1
$$
$$
\...
- 199
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1
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285
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Explicit examples of (probability) measures on $\prod \mathbb{R}$
Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
- 5,405
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3
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1k
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Reference request: book on stochastic calculus (not finance)
I am looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what ...
- 101
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463
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Expected number of connected components as $V(G)$ grows large
Let $E^c_n$ be the expected number of connected components of a simple undirected graph on the vertex set $\{1,\ldots,n\}$. (Every possible edge in $\big\{\{a, b\}: a, b\in \{1,\ldots,n\} \land a \neq ...
- 48.9k
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639
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Non-smooth Ito lemma for semi-martingales
Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any ...
- 5,405
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558
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Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]
I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying
$$\int_{\mathbb R}xd\...
- 1,835
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163
views
$\int_0^t f(s)\,dB_s$ normally distributed, mean and variance
Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$
How do I see that $Z_t$ is normally distributed?
What is the mean and variance?
I need ...
- 43
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1
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557
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Ergodic and mixing processes [closed]
I am working with an article, where it says:
"that the discrete time stationary sequence $\{Y_j\}_{j\in Z}$ is
mixing and hence ergodic."
where $Y_t$ is defined as
$Y_t = \int_{-\infty}^{t} h_k(...
- 3
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1
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258
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Are such averages known with representations of $S_n$?
Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to stick ...
- 1,893
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1
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370
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strong law of large number [closed]
Let $\{c_n\}$ be a descending sequence of positive real numbers, and let $\{X_i\}$ be a sequence of i.i.d. random variables.
Are the following statements equivalent?
$\operatorname{E}(X_1^2) < \...
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2
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435
views
conditional expectation under convex combinaison of probability measures(II)
Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...
- 1,835
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136
views
What is the area of the piece of an $n$-sphere within a given angle of a vector? [closed]
Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e.
$$ A(\theta) = \left\{ y \in \...
- 256
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1
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381
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Help prove a maximal inequality
Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number.
$S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$.
I want to prove:
$$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) \...
- 177
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1
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1k
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Generalizations of a product formula for the gamma function
Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
\...
- 265
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2
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294
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Relationship between these two probability mass functions.
If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows:
$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $
what can ...
- 141
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2k
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kalman filter: understanding the mathematical part
i am currently reading the Probabilistic robotics book where the filters are discussed.
Such filters as kalman filter or particle filters.
Now I can understand one thing while reading about the ...
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3
answers
770
views
Skewing the distribution of random values over a range
The following code comment comes from PHP, a free and open source project. I have done my own research and I cannot find any evidence to support the argument made in this code comment. Thus the ...
- 699
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307
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How to estimate the fraction of graphs with small clique among the graphs with certain edges
Among all $n$-vertex graphs with $M$ edges and constant $k$, how to estimate the fraction of graphs of clique less than $k$? Thanks.
- 3
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57
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Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 6,223
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89
views
Exchanging the integral and infimum on the space of couplings
Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on ...
- 305
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2
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135
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Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 150
0
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1
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159
views
Theories for "fuzzy" distributions
When calculating the probability density function for the quotients of adjacent values in an empirical time series, the image of the PDF looked like this:
It seems to resemble a lognormal ...
- 13.2k
0
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1
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182
views
Hidden Markov model with two hidden states?
I am currently studying what Markov models are, and have a question. If we have a hidden Markov model with 2 hidden states or observations, then how do we find the probability of just the main state ...
0
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1
answer
189
views
Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$
A family of function hash functions $\mathcal{H}:\{h:N\rightarrow M\}$ is call $k$-wise independent if whenenver $h$ is drawn uniformly from $\mathcal{H}$, let $x_1,\ldots,x_k$ be distinct elements of ...
- 121
0
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1
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107
views
Bounding $\|X_1/(X_1+X_2) - Y_1/(Y_1+Y_2)\|_p$ by the closeness of $X$ and $Y$
This question is inspired by the answer to this other question, but I have tried to make it self-contained and to zoom in on the counter-example from this answer.
Suppose $\{(X_n, Y_n)\}_{n=1}^2$ are ...
- 15
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1
answer
197
views
Bound the expectation of an average
Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
- 108
0
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1
answer
222
views
Condition for $f^\prime$ to be absolute integrable
Suppose $f(x)$ is the probability density function of a random variable $X$, which means:
$$\int_{a}^{b} f(x) dx = 1$$
Also suppose $f$ is continuous and differentiable.
Provide a non-trivial ...
- 143
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1
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124
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What is the probability space corresponding to the probability measure $\mathbb{P}_{p}$ in the context of this paper?
Here is the definition of the frog model we are interested in:
"... consider the homogeneous tree $\mathbb{T}_{d}$, that is, the rooted tree in which each
vertex has (is connected by edges to) $d ...
- 161
0
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1
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148
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Is the departure process of an infinite server queue independent of the arrival process?
Assume we have a $M/M/\infty$ queue with arrival rate $\lambda$ and a service rate $\mu$. From Burke's theorem, the departure process of the queue is a Poisson process with rate $\lambda$.
However, ...
- 13
0
votes
1
answer
105
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Transforming two smooth densities to the same density
I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
- 5