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

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4 views

Is Gaussian (X_1, X_2) optimal for $h(a_1X_1+ a_2X_2+Z_1) - \mu h( b_1X_1+b_2X_2+ Z_2)$?

Let \begin{align} W &= h(X_1+Z_1) - \mu h( X_2+ Z_2) \quad (1) \end{align} where $h(\cdot)$ is the differential entropy function, $\mu\ge 1 $ is a scalar, and $Z_1$ and $Z_2$ are independent ...
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0answers
17 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
0
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0answers
11 views

zero drift brownian motions and barriers problem [duplicate]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > ...
2
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0answers
18 views

Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace ...
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0answers
14 views

Suggestions for dealing with these order statistics

Consider a collection of $n$ random variables $X_i \sim N(\mu, \sigma^2)$, ($i = 1,2,\ldots, n$) and a random variable $X \sim \text{Exp}(\lambda)$. All $X_i$'s and $X$ are mutually independent. Let ...
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2answers
57 views

Antiderivative of $xe^{-cx^2}$

I need to define $c$ in $$\int_0^\infty xe^{-cx^2},$$ so that it becomes a probability-mass function (so that it equals 1). Where do I even begin finding the antiderivative of this? I know the answer ...
2
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1answer
15 views

How to find data distribution law using MATLAB?

Having a random variable $T \geq 0$ and a set of discrete data represented by $t=t_i$ and $P(T \leq t-i)$. My aim is to find the distribution law of $T$. Is there any fast method in Matlab that can ...
2
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1answer
20 views

Derive probability mass function from probability-generating function

Given the probability generating function $$G(z) = \frac{1}{2} \frac{3+z}{3-z}$$, how can one derive the pmf? I know that I have the manipulate the function into a series: $$G(z) = ...
2
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0answers
23 views

Permutation and combinations (Sheldon M.Ross)

We need to divide 8 new teachers among 4 schools , how many such divisions are possible ? I tried to solve this by the Distribution Method , that is : $x_1$ + $x_2$ + $x_3$ + $x_4$ = 8 , which ...
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1answer
22 views

determining distribution composed of uniform distributions

Let $X,Y,Z$ be i.i.d. $U(0,1)$ distributed. How can I determine the distribution of $$ \frac{X}{X+Y+Z}?$$ I have no idea how to go about this problem. Obviously this expression also has values ...
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0answers
10 views

Product of CDF and CCDF (or survival function)

Suppose we have two independent random Gaussian-distributed variables X and Y. X and Y represent thresholds for activation and deactivation, respectively. I'm interested in ensemble averaging over ...
1
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1answer
17 views

Multidimensional convergence in probability

If I have a vector $X^n=(X^n_1,...,X^n_m)$ is it true that $ \mathbb{P}(X^n\geq\epsilon)\rightarrow 0$ if $ \mathbb{P}(X^n_i\geq\epsilon_i)\rightarrow 0\ \forall i =1,...,m$ As $n\rightarrow \infty$?
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1answer
7 views

Terminology for probability matrix.

I have two related questions about terminology. If a matrix contains probabilities such that each column (or row or both) sums to $1$ , is this matrix always called a stochastic matrix i.e. even if ...
0
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1answer
19 views

How do I find the marginal probability density function when the interval is dependent of one of the variables?

I'm trying to find $f_x$ and $f_y$ given a joint probability distribution $$f(x,y) = \frac18 (y^2 -x^2)e^{-y}$$ defined on the interval $0 \leq y \leq \infty$, $-y \leq x \leq y$ Naturally I've tried ...
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1answer
26 views

Cumulative distribution function of the ratio of the maximum and minimum of two random variables [on hold]

Let $X_{1}$ and $X_{2}$ be independent, absolutely continuous random variables, each uniformly distributed between 0 and 1. I want to find the cumulative distribution function of the random variable ...
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0answers
10 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
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3answers
33 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
1
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0answers
22 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
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1answer
12 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
2
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1answer
30 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
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2answers
36 views

Integration limits of a Marginal Probability Density Function with a Triangle-Shaped Boundary

I have given a triangle shaped boundary $M$ of my probability density function in $\mathrm{R}^{2}$, with the limitations beeing: $$y = 0$$ $$y = x$$ $$y = 2-x$$ and a probability density function $$ ...
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0answers
40 views

Find the density of a ratio of random variables

$X$ has density $2x, 0 < x < 1,$ and $Y$ has density $1/10$ over $0 < y < 10$. $X$ and $Y$ are independent. I have to find (a) density of $Y/X$ (b) $E[Y/X]$ (c) $E[Y^2/X]$ I let $Z=Y/X,$ ...
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0answers
26 views

What is the name of a “Bernoulli” distribution with values in $\{1,2\}$ instead of $\{0,1\}$?

I have a "Bernoulli" distribution, but instead of $X$ taking values in $\{0,1\}$, it takes values in $\{1,2\}$. So $Pr(x=1)=p$ and $Pr(x=2)=1-p$. Is there a specific name for this distribution? ...
1
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1answer
103 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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1answer
11 views

Multivariate Quantiles

I am interested whether a concept for the multivariate equivalent to quantiles exists. In the univariate case, a $p$-quantile can be computed via the inverse of the cumulative density function, ...
1
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0answers
38 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
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1answer
12 views

Calculate Probability of a Range of Dice Rolls given their Distribution

I'll prefix this with - I'm not particularly great at Maths, so I might ask for an explanation of some of the answers. What I'm trying to do is convert this into something I can code. I've got a ...
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1answer
24 views

What will be the pdf from Mixture of Gaussians

In Euclidean $R^M$ space, I want to compute the pdf of the Euclidean distances between $d^M(\mathbf{z_i}) $= $||\mathbf{z_i -z_j}||^M = r_i^M , i \neq j$. What will be the pdf $f(r)$ ? Let there be ...
2
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1answer
27 views

Convergence of probability for $t$-distribution

Assume that $Z_0,Z_1,Z_2,\dots$ are i.i.d. RVs, $Z_j\sim N(0,1)$, and set $$T_n:=\frac{Z_0}{\sqrt{\frac1n(Z_1^2+\cdots+Z_n^2)}}$$ (a) Compute the limit $$\lim_{n\to\infty}\text{P}(T_n^2+2T_n\leq ...
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0answers
10 views

Poisson Distribution Optimization Problem

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
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0answers
206 views

Pareto distribution,fisher information, confidence interval [on hold]

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
1
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1answer
71 views

Density functions and estimators

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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1answer
26 views

upper bound and a lower bound on the number of points that are uniformly distributed on a surface

Can I calculate an upper bound and a lower bound (or max or min) on the number of points that are uniformly distributed on a surface, knowing the area of the surface ? More precisely, I have a sector ...
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0answers
15 views

Can we calculate the derivative of a distribution function with respect to its parameters?

I am asking a very basic question. Can we calculate the derivative of a density function with respect to its parameters, mean and variance? Can we calculate the derivative of a distribution function ...
1
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1answer
80 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
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0answers
28 views

References for the following functional

In many of the types of problems Ive looked at the following quantity keeps arising and I was wondering if anyone knew any references I could look at to learn some its properties. Take any function ...
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0answers
1k views

Sum of F Ratio distributed random variables

Where $X$ follows an F Ratio distribution F$(1,\alpha)$ with pdf: $$ f(x)= \frac{\alpha ^{\alpha /2} (\alpha +x)^{\frac{1}{2} (-\alpha -1)}}{\sqrt{x} B\left(\frac{1}{2},\frac{\alpha }{2}\right)},\; ...
1
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1answer
32 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...
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1answer
49 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
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0answers
6 views

Inverse Gaussian versus inverse Normal distribution [on hold]

I am wondering what is the difference between the inverse Normal and the inverse Gaussian distribution?
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2answers
20 views

CDF of a Uniform probability density function

I want to find Cumulative distribution function (CDF) of the following density function: $ f(x)= \begin{cases} 3/20 & \text{for } 2 \leq x \leq 4 \\[8pt] 4/20 & \text{for }4 < x \leq ...
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0answers
16 views

Distribution formed by taking two random points on an open disc and graphing their midpoint

I am wondering about the following distribution: Take an open disc and choose two points at random. and then take its midpoint in the new distribution. What does the resulting distribution look like? ...
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62 views
+50

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
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0answers
16 views

Joint cumulative density function of two independent Gaussian random variables

Assume we have two independent random variables $\theta_1$ and $\theta_2$ which each have separate Gaussian distribution functions $D_{\theta_1}$ and $D_{\theta_2}$. $\theta_1$ describes a threshold ...
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2answers
50 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
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0answers
25 views

Find limiting distribution

The question is like this: $X_i$ are i.i.d with $P(X_i\leq x)=1-e^{-x}$. $S_n=X_1+\cdots+X_n$. Find the limiting distribution of $\sum_{i=1}^nI(X_iS_n>1)$. It seems that the problem is related ...
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3answers
53 views

If $X$ has a Poisson distribution with $E[X]=\lambda$, does $Var[X^2]=4\lambda^3+6\lambda^2+\lambda$?

Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I ...
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0answers
18 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
5
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1answer
21 views

Density of stochastic integral

I am working on finding the PDF of $X_t^2$, where $X_t = \int_0^t A(u) \,dW_u$, a Wiener integral, i.e., $W_t$ is Brownian motion and $A(t)$ is a deterministic function. Here, would like to ask that ...
0
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0answers
30 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...