Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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0answers
23 views

What is the relation between the boltzmann distribution and Boltzmann equation?

The Boltzmann equation without collision operator $\Omega$ is as follows: $$\dfrac{\partial f}{\partial t} + \mathbf v \cdot \nabla f = 0 \quad\quad\quad\quad\quad(1)$$ Where $\mathbf v$ is the ...
-2
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0answers
7 views

Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
2
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1answer
84 views
+50

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
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0answers
18 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
0
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0answers
16 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
0
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1answer
5k views

How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
1
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0answers
29 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
1
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1answer
667 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
3
votes
2answers
2k views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
2
votes
1answer
18 views

An issue with the distribution function

I am reading a book about Boltzmann equation, here is a quotation: For a gas of $N$ particles, the number of particles having velocities in the $x$ direction between $c_x$ and $c_x + \mathrm dc_x$...
0
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2answers
21 views

Conditional Probability: Birth rank of children in randomly chosen families

(BH 4.7) A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
1
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1answer
52 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
0
votes
1answer
34 views

5 independent traffic lights, how many is car expected to pass without getting stopped

$\newcommand{\E}{\mathbb{E}}$ I can't wrap my mind around this one. I keep thinking it is geometric probability problem, but can't get correct solution (which is $\E(X) = 0.6598)$. Problem : ...
1
vote
1answer
44 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
7
votes
1answer
197 views

Is there any probability model for multi-stage motion of an object such as this.

I have this following case (please refer to attached pic below) where a particle is resting on the ground and it needs a minimum amount of force (Fmin) to reach from one level to the next level. But ...
2
votes
0answers
36 views
+150

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
0
votes
0answers
20 views

Is a particular sample a 'random sample'?

Say I have a high dimensional Bernoulli distribution $X$ (defined by $p_1,p_2,...,p_n$, independent marginals). I realized I cannot use the chi-squared test to check if a given set of $m$ samples from ...
0
votes
1answer
23 views

Prove $X$ and $Y$ are not independent

Let $X$ and $Y$ be two random variables. Their joint probability density function is $$f: (x, y) \mapsto C(y^2 - x^2)e^{-y} \mathbf{1}_A(x, y)$$ where $C \in \mathbb{R}$, $A = \{(x,y) \in \mathbb{R},...
1
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0answers
18 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
5
votes
5answers
19k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
1
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2answers
54 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
2
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1answer
19 views

Distribution of the minimum

I have the following problem, given a random variable $X$ with density $$f(x)=2x\text{ for }x\in(0,1)$$ and a r.s.s. $X_1, X_2, X_3$. I have to calculate the probability that $X_{(1)}=\min\{X_1,X_2,...
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0answers
19 views

Communicating classes of a power of the irreducible transition matrix? [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P^k$. In terms of $d$ and $k$, how many communicating classes does $P^k$ have, and what is the period ...
1
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0answers
34 views

Multivariate to univariate distribution

Say one has a Student's t-copula (where all the margins and the copula can have different degrees of freedom). If you had a matrix of data (f.e. financial returns) and you know that the portfolio ...
0
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0answers
31 views

would it be possible to measure neural network combination/ representation as capacity?

My question is about machine learning, If you are familiar with you the restricted Bolatzmanne machine. I would like to know if we can implements some ideas. RBM in simpler case is just two layers ...
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votes
0answers
13 views

Simple Markov property on stopping times

Suppose $(S_n)_{n\geq1}$ is a Markov chain on the two dimensional lattice of the integers. Then define the stopping time $\tau_A'=\inf\{n\geq1:S_n\notin A'\}$ and consider the following for $A\subset ...
1
vote
1answer
27 views

Transformation technique to find PDF

Consider two random variables with the following joint PDF: $$ f_{X,Y}(x,y) = \begin{cases} 2, & x > 0, y > 0, x + y < 1 \\ 0, & \text{otherwise} \end{cases} $$ I need to find ...
1
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1answer
650 views

Probability generating function for logarithmic series distribution, support $k\geq1$

I'm trying to derive the probability generating function (pgf) for the logarithmic series distribution, and not getting the expected form $\frac{\log{(1-qs)}}{\log{(1-q)}}$. It seems that pgfs are ...
0
votes
3answers
40 views

Generating samples from a Beta(2,2) distribution

I'm looking for a convenient way to generate $\text{Beta}(2,2)$ random variables, using independent $\text{Uniform}(0,1)$ random variables and elementary functions. I'd prefer to avoid rejection or ...
3
votes
1answer
28 views

Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
2
votes
1answer
27 views

Probability distribution of order statistics

Let $X_1$, $X_2$ and $X_3$ be independent random variable with continuous distribution $$f(x;\theta)=\frac{1}{\theta}I_{(0,\theta]}(x), \ \theta \gt 0$$ I need to find distribution of $Z=\frac{X_{(...
0
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1answer
27 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [on hold]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
0
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0answers
17 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
2
votes
1answer
179 views

Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$

$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ? I ...
1
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1answer
26 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
3
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1answer
540 views

Fast generation of Pareto-distributed randoms.

I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
0
votes
1answer
12 views

Distribution of binomial with poisson-distributed trials

Assume there is an underlying process, $K$, governed by a Poisson distribution, $K\sim Poisson(\lambda)$. We observe realizations this process, but with an imperfect count detection device with ...
-1
votes
2answers
42 views

Probability of even sum of $n$ integers with uniform distribution from $\{1,2,\dots, 2n\}$.

Choosing with Uniform distribution $n$ numbers from $\{1,\dots,2n\}$ with returns and the order is important. What is the probability that the sum of these number will be even? Thanks.
1
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0answers
34 views

Equivalent definitions for weak/distribution convergence

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
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0answers
17 views

find the conditional function (continuous- discrete)

Let $X_1$ be a continuous random variable with pdf $f(x)=\frac{2}{x^3},~~~~x>1$. $0 $ otherwise. Additionally the random variable $X_2$ is defined as $X_2=I_{[1,2]}(x)$. ($I$ := Indicator function) ...
3
votes
1answer
65 views

How to evaluate this Fourier Transform $A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$

This is basically the Fourier transform of a Student´s T pdf. How do we compute it? $$A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$$ for $\nu$ any number greater than zero ...
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0answers
22 views

Fisher information for exponential family: Regularity conditions

for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus....
3
votes
1answer
716 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
0
votes
0answers
32 views

Function of random variable: Two ways to find the pdf

Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function. Method I first using the ...
4
votes
1answer
49 views

Compute the covariance of $\xi$ and $\min \{\xi,2\}$, where $\xi$ is exponentially distributed

Let $\xi$ be a random variable exponentially distributed and let $\xi_1=\min \{\xi,2\}$. Calculate $Cov(\xi,\xi_1)$. I know the problem is easy but I just need somebody to check my work. Here's my ...
1
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1answer
786 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 \...
4
votes
1answer
40 views

Why is the domain of the error function scaled by $\sqrt{2}$

The normal distribution function $\Phi(z)$ has the definition $\Phi(z) \equiv \frac{1}{\sqrt{2 \pi}} \int_0^z e^{\frac{-x^2}{2}} \, dx$. However the error function is conventionally defined such that ...