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

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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 ...
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2answers
20 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 ...
2
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1answer
652 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
2
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1answer
42 views

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose variance is distributed Log ...
2
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1answer
2k views

The math behind flappy bird

You may have heard of this game called flappy bird, but even if you haven't, you should be able to understand this basic game: The player progresses through a series of obstacles. The probability of ...
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1answer
44 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 ...
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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 < ...
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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 ...
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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 ...
4
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1answer
32 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. ...
2
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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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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
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
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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 ...
0
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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
51 views

In the space of probability distributions, is the set of discrete distributions dense?

Is the following true: In the space of probability distributions, the set of discrete distributions is dense with regard to the Lévy metric. Can someone point me to any reference on this ...
2
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3answers
717 views

Suppose 5 red and 7 green balls are in a bag. Three balls are removed without replacement.

Suppose $5$ red and $7$ green balls are in a bag. Three balls are removed without replacement. What is the probability that the second and third balls are both green? I'm having trouble figuring out ...
2
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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 ...
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1answer
176 views

Joint distribution of an infinite collection of random variables?

Let's say we have a countable collection of random variables $X_1, X_2, ...$, in $(\Omega, \mathscr{F}, \mathbb{P})$ Can we define a joint distribution function for all of them ie $$F_{X_1,X_2, ...
0
votes
1answer
22 views

Possible off-by one error in a solution of a geometric distribution problem: $X\ge 5$ instead of $X\ge 6$

The Problem The problem I am having trouble with is as follows: An urn contains five red balls, three white balls, six black balls, and six blue balls. Assume that the sample is drawn ...
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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 ...
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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, ...
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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 ...
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2answers
421 views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
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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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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 ...
2
votes
1answer
52 views

Vandermonde's identity and the close form of $\sum_{k=0}^r C(n,k) C(m,r-k) x^k$

I have a question related to Vandermonde's identity: From Vandermonde's identity, we have: $$ \binom{n+m}{r}=\sum_{k=0}^r \binom{n}{k}\binom{m}{r-k} $$ Now, I have an extra term $x^k$ inside the sum, ...
1
vote
1answer
40 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
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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 ...
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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 ...
0
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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, ...
0
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1answer
54 views

Distribution of the maximum of covariant random variables

I am looking to determine the distribution of $\max(X_i,Y_i)$ where $X_i = |A_i|^2$ $Y_i = \frac{1}{2}|A_i - A_{i-1}|^2$ Here $A_i$ is a complex vector with normally distributed real and imaginary ...
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1answer
1k views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, ...
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2answers
51 views

Bayes Formula for a problem?

So I am taking exams in Probabilities next week and this one comes from last year's exams period. The problem is the following. A system of 5 quantities is functional if and only if at least one ...
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0answers
47 views

The higher moments of truncated Gaussian

We assume that $$X \sim N(0,1/d),$$ where $d\rightarrow \infty$. For $\delta > 0$ sufficiently small. My question is, what is the correct order (in terms of $d$) of $$ ...
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0answers
20 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
0
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0answers
29 views

Trouble generating custom probability density function

I want to generate random distributed numbers from a uniform distribution ($x$ is a uniform distributed number). The probability density that I want to obtain is: $$ f(y) = \mathcal{N}e^{\beta y} $$ ...
3
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2answers
2k views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
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0answers
27 views

How can I compute the number of selected green ball from a given selection prob.

I have $2$ red balls in box 1 and $4$ green balls in box 2 as figure. The prob. selection the red balls (R) in box 1 is $$P(R=1)=0.1$$ $$P(R=2)=0.9$$ And prob. selection the green balls (G) in box ...
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5answers
10k views

What is the use of moments in statistics

Can any one give an "simple" explaination about what is the use of moments in statistics.Why we need moments? what we can learn from it? if possible please use less equations. Advance thanks for your ...
2
votes
0answers
36 views

Compound Distribution — Log Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (mean of the log of ...
3
votes
0answers
57 views

Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean ...
0
votes
1answer
24 views

Joint CDF's of both continuous and discrete random variables

I am working on some homework and have arrived at a problem that has me stumped. I am trying to find the conditional probability of Y given a discrete variable X or: $$F_{Y|X}(y)$$ which I know is ...
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0answers
24 views

Could you please explain how do we get the second term while computing $E[f(Y)]$

I will fix the dimension $n$ and use $S:=\{x\in\Bbb R^n:\|x\|=1\}$ to denote the unit sphere. Let $\sigma$ denote surface measure on $S$, and define $\bar\sigma:=[\sigma(S)]^{-1}\sigma$, the "uniform ...
0
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1answer
30 views

Mean number of unique choices given n people choosing randomly from a set of N elements

My question is similar to the birthday problem, but I can't seem to find a simple solution. The question (in a general form) is that, given a set of $n$ people who each choose elements from a set of ...
1
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1answer
899 views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
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0answers
23 views

Show the Statistic is Complete

Consider a random sample $Y_1,\ldots,Y_n$ of the Uniform Distribution on the Interval $[-\phi,\phi]$ I'm wondering how I can show that the Statistic $$ T(\mathbf{Y}) = ( Y_{(1)} , Y_{(n)}) $$ is a ...
1
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1answer
30 views

Limit of Snedecor's F

Suppose we have a random variable $X$ such that $X\sim \dfrac{d}{n-d}F(d,n-d)$, with $d,n\in\mathbb{Z}$. What happens when $n\to\infty$? And when $d\to\infty$? I think when $n\to\infty$ then it goes ...
1
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2answers
45 views

Given two exponentially distributed random variables, show their sum is also exponentially distributed

Given two independent exponentially distributed random variables, I want to show their sum is also exponentially distributed. This is my try, I used convolution. It didn't get me anywhere...
0
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0answers
14 views

SD probability problem involving chi-square distribution

Preparing for an exam in statistics, I have been pondering the following problem: Given that in country X 14 % of people hold a university degree, find the probability that for random sample of size ...