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

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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
vote
2answers
49 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
votes
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,...
-1
votes
0answers
14 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
vote
0answers
21 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 ...
1
vote
1answer
31 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 \...
0
votes
0answers
26 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 ...
0
votes
0answers
7 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
23 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
vote
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
35 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 ...
0
votes
2answers
17 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 ...
6
votes
1answer
136 views
+200

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 ...
1
vote
1answer
48 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 ...
3
votes
1answer
27 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
21 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_{(...
2
votes
0answers
24 views

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 $...
-2
votes
0answers
17 views
0
votes
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
votes
0answers
16 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
177 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
vote
1answer
25 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
votes
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
11 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
33 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.
0
votes
0answers
42 views

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 $...
1
vote
0answers
30 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 ...
1
vote
0answers
15 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
61 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 ...
0
votes
0answers
18 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
vote
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 ...
-2
votes
0answers
17 views

Probability calculation [on hold]

There are n number of data. n data contain x ideal data and y raw data. Question: how to select randomly one ideal data. Please give me calculation with explanation. Thanks in Advance
0
votes
2answers
51 views

Find the probability of selecting exactly $14$ defective items.

$70\%$ of items are defective. You randomly select $20$ items. Find the probability that the number of defective items is exactly $14$. I have $n$ as $20$, $x$ as $14$, $p$ as $.7$ and $q$ as $.3$. ...
0
votes
1answer
25 views

Expected value and variance of discrete random variable

Let $Y$ be a discrete random variable with density function: $$p(y;\theta)=\left(\frac{\theta}{2}\right)^{\lvert y\rvert}(1-\theta)^{1-\lvert y\rvert}$$ where $y\in\{-1,0,1\}$ and $\theta \in[0,1]$...
0
votes
0answers
16 views

Indicator variable syntax

Are these syntax equivalent? $$f(x,\lambda)=\lambda e^{-\lambda x}I_{(0,+\infty)}(x),\ \lambda > 0$$ $$f(x,\lambda) = \left \{ \begin{array}{cl} \lambda e^{-\lambda x} & x \gt 0 ...
0
votes
0answers
13 views

Determine eigenvalue distribution support

I am working on a project regarding random matrix spectra and I need some help with the following: let us assume we are looking at some particular family of NxN random matrices in the limit of N -> ...
0
votes
2answers
81 views

A fair die is rolled 100 times. Which of the following has a probability of at least 95%?

A fair die is rolled 100 times. Which of the following has a probability of at least 95%? $ $ 1.) Sum of the rolls is greater than 322 2.) Sum of the rolls is less than 392 3.) Number of rolls ...
3
votes
0answers
44 views

How to approximate the cumulative distribution function of the normal by a product of functions?

Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{...
3
votes
1answer
272 views

Stationary distribution of a vector-autoregressive process

Given a $K\times K$ real matrix $\mathbf{\Phi}$ and given a sequence $\boldsymbol\varepsilon_t$ of multivariate normal variables $\boldsymbol\varepsilon_t\sim \textrm{N}\left(0,\mathbf{\Sigma}\right)$,...
0
votes
0answers
12 views

find out min/max of statistical distribution (GPA) from median, mode, count, size of elements? [on hold]

I would like to find to the min/max of a distribution given the following. Was wondering if it is possible. You could think of them as GPA Number of elements in distribution: 93 Theoretical min of ...
0
votes
0answers
38 views

Central Limit Theorem for gambling return ratio

Consider a single bet with odds $o$ and thereby implied probability $1/o$. Assume that the real probability $p$ is known. Let $I$ be the stake, and $y$ the return from the bet. Then, $\mathbb{E}(y) ...
2
votes
1answer
24 views

Confusion in Calculating Conditional Probability mass function

Question: If $X_1$ and $X_2$ are independent binomial random variables with respective parameters $(n_1,p)$ and $(n_2,p)$, calculate the conditional probability mass function of $...
0
votes
1answer
20 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
0
votes
0answers
15 views

Queue depth to keep workers busy

I'm trying to find a probability of keeping w workers busy with a q queue depth feeding those w workers. When the queue has at least one item in it the item can be taken and the item was randomly ...