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

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0
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14 views

How do I find the marginal distribution with this summation/series?

Let $X$ be a Poisson(2) and $Y$ be Binomial(10,3/4) random variables, If $X$ and $Y$ are independent, then $P(XY=0)$ is I thought of using transformation to find the distribution of $XY$ so I let ...
0
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1answer
28 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
2
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0answers
39 views

How to derive the distribution of a variable linearly related to two others?

Say i have $$x= \beta + \alpha \ln(y) + \varepsilon, $$ where $E(\ln(y)) = 0$, so $E(m) = \beta$, and $\varepsilon$~ $N(0, \sigma^2)$. Assume $$f(y) = \tau f^A(\ln(y)) + (1 - ...
4
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1answer
74 views

From marginal distribution to joint distribution

Consider two sequences of real-valued random variables, $\{X_n\}_n$ and $\{T_n\}_n$. Let $\rightarrow_d$ denote convergence in distribution. Assume (1) $X_n\rightarrow_d L$ as $n\rightarrow \infty$, ...
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0answers
6 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
1
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1answer
45 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
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0answers
23 views

Where does the name of the hypergeometric distribution come from?

I understand what it does and how to get there, but why is it called hypergeometric? All the other distributions I know of have rather self-explanatory names like "binomial" or "exponential", or are ...
2
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0answers
56 views

Sampling from a given pdf

I have the following pdf: $$ f(x) = C x^d I_0\left(b \sqrt{- \log\left(\frac{x}{A}\right)}\right)$$ for $0 < x \leq A$, $C$ is a normalizing constant, $b$, $d$ are constants, and $I_0$ is the ...
2
votes
2answers
25 views

cumulative distribution of intersection of events

Let $X_1,\dotsc,X_n$ be independent identically distributed random variables having common distribution function $F_X(\cdot)$. Express the event 'the smallest of the $X$s exceeds $k$' as an ...
2
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1answer
35 views

Are squares of two i.i.d normal distributions i.i.d?

Let $X,Y\sim N(0,\sigma^2)$ i.i.d. Correct me if I'm wrong, but $X^2,Y^2$ are also I.i.d. Could I get advice on how to show this, or could someone let me know if this is difficult to show? My ...
1
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0answers
18 views

Probability in a Dice Game [duplicate]

By the following problem: Two players, $A$ and $B$, play a game where they, one after another, throw two dices. The first one to get a sum of $A$ wins the game. Let player $A$ be the one to play ...
1
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1answer
25 views

Asymptotic distribution of a measure of homogeneity

For an exam preparation I'm trying to solve the following question, but I get stuck. The question is One measure of the homogeneity of a multinomial population with $k$ cells and probabilities ...
0
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0answers
29 views

Getting the independent variables from dependent variables. [duplicate]

This question is related to the solution in the answer here: Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. Quick description of my problem: Let ...
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0answers
13 views

Comparing distributions with continuous parameter

Suppose we have a set of distribution $P(n; T)$ which depends continuously on a real parameter $T$. We want to determine what the smallest change of $T$ is such that the distribution becomes ...
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0answers
23 views

Limiting Distribution of what was Chi square

Let $Z_n\sim \chi^2(n)$ and $W_n= Z_n/n^2$. Find the limiting distribution of $W_n$. Using the delta method seems a bit unusual, since the function in question would be $g(z) = z/n^2$. Thus, I ...
0
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0answers
28 views

Very (positively skewed distribution?

Could anyone give me an example of a sequence of cumulative distribution functions $F_i$ such that $F_i(\mu_i)\to 0$ as $i\to \infty$, where $\mu_i$ is the expected value of $F_i$? It would be better ...
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0answers
11 views

Normalized multivariate random vectors

While it is well-known that normalizing any zero-mean elliptically symmetric multivariate vector, such as a multivariate Gaussian, results in angular Gaussian distribution, what happens if the ...
2
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0answers
100 views

Probability mass function of the sum of the function of the sum of iid random variables

How can I get an expression of the probability mass function of: \begin{equation} Y_i=\sum_{k=1}^i f\left(\sum_{n=1}^{k} X_n\right) \end{equation} being $x_n, n=1,2,...$ iid random variables and ...
2
votes
1answer
47 views

Expectation of a function of a normally distributed random variable

Consider that I have to produce this result: $$E[u(W_0+r(\theta))] = u(W_0)+\theta-\frac 12\rho\sigma^2$$ From this: $$ E[u(W_0+r(\theta))] = \int_{-\infty}^\infty u(w_0+r) \frac{1}{\sigma ...
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0answers
14 views

Deriving Time of extintion of a Small neural Network

I'm trying to derive the Expected Value of the Time of Extintion $\tau_{ext}$ of a small Neural Stochastic Network with the following dynamics, where I consider $\tau_{ext}$ to be the time of the last ...
2
votes
1answer
32 views

Distribution of a random variable waiting for a consecutive sequence of bits?

Suppose we're trying to transmit a message comprised of $n$ bits. Assume each bit has a probability $p$ of being correct. Success means we succeed at consecutively transmitting all $n$ bits. As soon ...
2
votes
1answer
47 views

How to pick the same color spheres out of two boxes

For the following problem: There are two boxes $A$ and $B$. Box $A$ contains $3$ red, $8$ white and $13$ green spheres, while box $B$ contains $5$ red, $7$ white and $6$ green spheres. If we pick ...
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0answers
8 views

fidi of Chi-square Random Field

The $\chi^2$ random field $U(t)$ with $n$ degree of freedom (dof) is defined as: \begin{align} U(t) = \sum_{i=1}^n X_i(t)^2, t\in\mathbb{R}^N \end{align} where $X_1(t),...,X_n(t)$ are i.i.d ...
1
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1answer
29 views

Probability that a random variable is zero as expressed as limit of a sequence

Consider a random variable $U:\Omega \rightarrow \mathbb{R}$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $U(\omega) \geq 0$ $\forall \omega \in \Omega$. Consider a ...
0
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2answers
21 views

convergence in distribution of truncated gaussian variables

Let $X$ be a random variable which is distributed normally with mean $\mu=0$ and variance $\sigma=1$. Suppose that $X_n$ is a random variable for any positive integer $n$ with truncated normal ...
0
votes
2answers
57 views

Finding marginal distribution, unit sphere

I'm asked to find the marginal distribution of $(X,Y)$ as $(X,Y,Z)$ is a point chosen uniformly on the unit sphere. I've worked out that the joint density function $f_{XYZ}(x,y,z) = \frac{3}{4\pi}$ ...
0
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2answers
46 views

The distribution of the x-coordinate on unit circle

I'm trying to determine the distribution of the x-coordinate (uniformly distributed) on the unit circle (density function). I've seen some threads on this, such as this, where they use the method of ...
0
votes
1answer
20 views

Marginal distribution for a set of discrete events in continuous time

Assume we have a set of four events $\{(k_1,t_1),...,(k_4,t_4)\}$ where the $k_i$ label the type of event from a finite set of possible events, and the $t_i$ their respective times, with $t_i < ...
1
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1answer
18 views

Determine expected lifetime of a TV purchased from 1 of 3 factories, using LIE.

Problem Statement: Consider a TV made in one of three factories, namely, A,B, and C. Note that the quality of work done at each factory is different. The pdf of time to failure, X, is given as ...
1
vote
1answer
45 views

Probability of lim sup, lim inf for sequence of random variables.

Maybe this is extremely simple, but i havent found a specific answer for this online. For a sequence of independent continuous random variables $X_n$ ,$n=1,2,3,...$ , all with the same probability ...
0
votes
1answer
28 views

Computation of a joint distribution function

Let $Y = X+W$ and suppose the joint PDF of $X$ and $Y$ is $$ f_{X,Y}(x,y) = \lambda^{2}e^{-\lambda\cdot y} \hspace{2mm}:\hspace{2mm} 0 < x < y < \infty$$ What is the density of $W$? I have ...
2
votes
1answer
42 views

What do the square bracket signify in $\int [\text{d}\pi]f(\pi)$

I am reading this paper which repeatedly includes integrals such as, $$ P_M(\phi \to \phi') = \int [\text{d}\pi][\text{d}\pi'] P_G(\pi)\delta((\phi, \pi) - (\phi'', \pi'')) $$ Note ...
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0answers
14 views

Problems about Farlie-Morgenstern family of bivariate CDFs

Hi I am trying to solve the following problem: Let $F_X:\mathbb{R}\to[0,1]$ and $F_Y:\mathbb{R}\to[0,1]$ be unnivariate Cumulative Distribution Functions (CDFs) and suppose $-1\le\alpha\le 1$. Define ...
4
votes
1answer
33 views

Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper: We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology. ...
0
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2answers
35 views

Conditional Probability of Two Poisson RV's

Question: During a given year for a circus performer, let X represent the number of minor accidents, and let Y represent the number of major accidents. The joint distribution is: $f(x,y) = \Large ...
1
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0answers
23 views

deriving joint density of $X, W$ while knowing joint density of $X, Y$ where $X + W = Y$

Joint density of $X,Y$ is $$f_{X,Y}(x,y) = a^2e^{-ay},$$ and I know that $X + W = Y$, where $0 < x < y < \infty$ I want to determine the joint density of $X$ and $W$. I know I can let $X = ...
0
votes
1answer
68 views

Is the expected value of the difference of these two random variables, with infinite expected value, $0$, or undefined?

Let's say we have two independent random variables, $x_1$ and $x_2$, both have a probability mass function $X$ defined as $$X(n) = \begin{cases} 2^{-m} & \text{if $n=2^m$ for $1 \le m \in \mathbb ...
1
vote
1answer
66 views

Proof that process is martingale, exponential distribution

Let $X_1,X_2,\dots$ be i.i.d. random variables with exponential distribution with parameter $1$ and define $$Y_m= \sup{\{k\ge1:X_1+\dots+X_k\le m\}}$$ Prove that $Y_m-m$ is martingale and ...
0
votes
1answer
34 views

PDF of negative $\cos(X)$

Let $Y = - \cos(X)$, then what will be the pdf? Please share if you have any idea. If $Y = \cos(X)$, where $X$ is uniformly distributed in the interval $(0, 2 \pi]$, then the pdf is given by ...
2
votes
1answer
26 views

Can you create non transitive dice for any finite graph?

Let's say you have a finite directed graph, with no two nodes that point at each other. Can we assign each node a dice, so that each node beats the node it is pointing at. This is easy for acyclic ...
0
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1answer
22 views

Find the value for z(0.1) from a distribution table?

I'm doing a statistics course, and I thought I had no problems using distribution tables to find values. For example, for the Gauss distribution, if I want $\Phi(-2)$, I will do $1 - \Phi(2)$ because ...
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0answers
19 views

Truncated Gumbel I extreme value probability density function (PDF)

Can anyone confirm the method is correct that I am using to get the truncated Gumbel I extreme value PDF. I am fitting Gumbel I to ice loads that occur on ships, if that is of any help here. The ...
0
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1answer
18 views

How to find the PDF for Y=|X-1|?

Given $X\sim N(0,1)$ and $Y=|X-1|$. Find the PDF of $Y$. I tried to discussed when $x>1$ and $x\le1$, but this gives me two different functions and I have no idea how to combine them. However, ...
0
votes
0answers
21 views

Geometry of Vector Random Variables and Joint Distribution

I'm not a statistician but have been trying to understand the following problem in my research: I have two $3\times 1$ random vectors $\mathbf{v}$ and $\mathbf{w}$, and a function ...
0
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2answers
21 views

What is wrong with my solution?

Given a person makes repeated attempts to destroy a target, attempts are made independent of each other. The probability of destroying the target in any attempt is $.8$. Given that he fails to ...
0
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0answers
28 views

How to estimate a distribution from samples in a histogram

Given a r.v. $\tau$ , I've computed $\Bbb{P}(\tau >a)=e^{-Nx}(e^{Nxe^{-a}}-1) $ , where $N\in\Bbb{N}_{>1} $ and $ x\in \Bbb{R}_{>0} $ are just fixed parameters; say $N = 2 $ and $ x = 1$, ...
0
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0answers
21 views

Implications of symmetric probability density function

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with probability density function $f$. Suppose $f$ is symmetric around zero. This ...
1
vote
1answer
17 views

A Game that follows Hypergeometric Distribution?

I have the following problem: Inside a box there are $2$ white and $3$ black spheres. Two friends, $A$ and $B$ play the following game: They pick, one after another, a sphere from the box without ...
0
votes
1answer
19 views

Given $(X,Y)$ a Gaussian random vector, find the properties of $(X-Y,X+Y)$

Given $f_{X,Y}(x,y)=\frac{1}{(2\pi)^{n/2}(\det \Sigma)^{1/2}}e^{-\frac{1}{2}(\vec x - \vec \mu)^{T}\Sigma^{-1}(\vec x - \vec\mu)}$, I want to find $\vec\mu$ and $\Sigma$ of $(U,V)=(X-Y,X+Y)$. First, ...
1
vote
1answer
31 views

Bhattacharya Distance on Distributions (Matrices) with Different Number of Variables (Dimensions)

We have two matrices, $A$ and $B$, representing two different probability distributions, with dimensions, $m*n$ and $k*n$, respectively. How can we calculate the Bhattacharya distance or another ...