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

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2answers
18 views

Question about finding a distribution without taking into account previous events

We have 8 prisoners, each has a probability of escaping (independently) each day of $0.4$, what is the distribution of the amount of escaping prisoners on the third day? This is the answer: the ...
2
votes
1answer
38 views

Conditional distribution of $X$ exponential given $U\leq e^{-X}$, with $U$ uniform on $(0,1)$

Let $X$ be exponentially distributed with mean $1$ and $U$ be a $U(0,1)$ random variable independent of $X$. Define $$I= \begin{cases}1,&U \leq e^{-X}\\ 0,&\text{ ...
0
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0answers
19 views

Probability distribution to failure [closed]

I am going to do a simulation for a manufactruing system, i must consider a scenario as: a $20\%$ probability of failures occurring in $M1$. Q: What is the probability distributions the time to ...
2
votes
2answers
56 views

Distribution of a convolution.

Assume that $X_1,X_2,X_3,X_4$ are IID such that $P(X_1=0)=0.3, P(X_1=1)=0.1$ and $X_1$ has on $(0,1)$ the density $f(x)=0.6$. Calculate $P(X_1+X_2+X_3+X_4 \leq 1).$ My work so far. It seems that ...
-1
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0answers
40 views

Finding distribution from PGF not in closed from.

$X_1,X_2,\ldots,X_N$ are independent and identically distributed random variables. We have $X = e^{-Y}$, where $Y\sim\mathrm{Poisson}(\lambda_u)$, and $$Z =X_1+X_2+\cdots+X_N ,$$ where $N \sim ...
1
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3answers
32 views

How can $e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$ given $X \thicksim N(\mu, \sigma^2)$ when they have different support?

According to Wikipedia (page about lognormal distribution), if $X \thicksim N(\mu, \sigma^2)$ then $Y=e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$. But the support of $\mathrm{logN}$ is just ...
0
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1answer
23 views

Recovering density parameters from distribution function

Let $X$ be a random variable with probability density function $g(x;\theta_1,\theta_2)$, where $g$ is parameterized by two real numbers $\theta_1$ and $\theta_2$. I'd like to specify that $$ P(a \leq ...
1
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0answers
76 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
0
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2answers
56 views

Random increment through a probability distribution function

To Clarify i am trying to generate a random variable from a gamma pdf If $\Delta X$ indicates a random increment and it is said that $\Delta X$ follows a Gamma distribution. What would that mean ...
2
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0answers
39 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
0
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0answers
33 views

Can I compute marginal distribution this way?

I have posted the same question in the Internet another website. But I did not get the answer replies. I only can come here to have a try. The math statement I put here may not be correct. You can ...
0
votes
1answer
34 views

Sum of two truncated normaly distributed variables

Let $X$ and $Y$ be two variables which are truncated normally distributed above zero (that is $X$ and $Y$ have the lower truncation point zero, their values are bounded above zero). Is $X+Y$ truncated ...
1
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1answer
18 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
3
votes
3answers
38 views

Probability of Punctures for a group of cyclists

The matter of the probability of punctures occurring cropped up during a ride yesterday with a friend. His view is this, (As we can't let a subject drop.... ;-) ) "Eric, There must be more chance ...
0
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0answers
33 views

Help with textbook formula

In Bishop - Pattern Recognition and Machine Learning, Section 1, I do not fully understand Formula (1.65). Although it's not stated explicitly, I assume that I is the identity matrix with the ...
1
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2answers
34 views

A smooth function satisfying these functional constraints

I am looking for any function on a square $$f:[-1,1]\times [-1,1] \rightarrow [0,1]$$ with the following properties: The function $f$ is as smooth as possible, e.g. differentiable almost everywhere. ...
2
votes
1answer
61 views

Transformation theorem

Given $X_1$ is $\Gamma(\alpha,1)$ distributed and $X_2$ is $\Gamma(\beta,1)$ distributed and set $$Y=\frac{X_1}{X_1+X_2}.$$ The task is to show that $Y$ is $\operatorname{Beta}(\alpha,\beta)$ ...
1
vote
1answer
47 views

Probability of an event that occur first of a joint uniform distribution

A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time ...
0
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0answers
13 views

4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...
0
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0answers
36 views

Topology of statistical manifolds

I am currently working with statistical manifolds. Roughly, a statistical manifold is a set of distribution parametrized by a set of parameters. However i have trouble finding more precise definition. ...
2
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1answer
18 views

The distributions of incomes in two cities follow the two Pareto type pdfs. Find P(X<Y)

The distributions of incomes in two cities follow the two Pareto type pdfs $$f(x)= \frac{2}{x^3}, 1 < x < \infty.$$ $$g(y) = \frac{3}{y^4}, 1<y<\infty.$$ Here one unit represents ...
1
vote
1answer
26 views

What is the time between groups of events when single events have a Poisson distribution?

I'll ask this with a concrete example to be clear. Let's say I have a Poisson process that tends to produce one event every two minutes. Then the probability of getting an event in a given minute is ...
1
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0answers
14 views

To what set do measured values belong?

This question is more conceptual than practical. It seems that when we apply mathematics to measured values, we treat them like real numbers. When measured values take error into account using ± ...
1
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3answers
35 views

Creating a PDF for a discrete random variable with a countably infinite set of values?

I am unsure how to transition from discrete random variables with a finite set of values to ones with a countably infinite set of values. The question that spawned this problem: A bucket has two ...
0
votes
1answer
34 views

How to reconstruct distribution from the generating function

Suppose $$F(x)=\sum_n p(n)x^n$$ is a generating function, and we have the expression for $F(x)$ explicitly. Then how we can get the expression for $p(n)$ from this generating function?
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0answers
33 views

difference between limiting and special case

In Mathematics and Statistics we see generalized distributions having a number of parameters. varying the values of these parameters we get special or limiting distributions of the generalized ...
1
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1answer
37 views

Generate random number according to any equation

So I'm after a random number generator where the probabilities of a number occurring in some range is matched to some function. Only really looking at functions with nice integrals (for simplicity ...
0
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2answers
49 views

If we know X is a Poisson binomial random variable what can we say about mX?

Suppose that X is sum of m independent Bernoulli random variables that are not necessarily identically distributed, and thus it has Poisson binomial distribution. Is mX also a Poisson binomial random ...
1
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2answers
33 views

Probability ( Random Variable ).

Let $$p_X(x) = \begin{cases}\frac{x}{15},& x\in\{1,2,3,4,5\}\\ 0,& \text{otherwise}\end{cases}$$ be the probability mass function of $X$. We need to find $$\mathbb ...
0
votes
1answer
43 views

Expectation on second moment which involves linearity

I have a small problem regarding to expectation on second moment. It would be lovely if you guys can give me a hand. The amount of a claim that a car insurance company pays out follows an ...
0
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1answer
67 views

Probability distribution of bored people

5 people are arranged in a row, a person is talkative with a probability of $p$ and silent with a probability of $1-p$, each is independent. A person is bored if he's talkative and sits between two ...
1
vote
0answers
32 views

Expectation of $\min(X, c)$ for $X$ truncated r.v. and $c$ constant

I have a random variable $X$ and a constant $c\geq 0$. I define the r.v. $Y = \min(X, c)$ and I want to calculate $E[Y]$. I have seen different posts on similar topics, so I am trying to pull all ...
1
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0answers
17 views

Is there a map that maps the mean vector and the variance matrix of a multivariate lognormal to its location vector and diffusion matrix?

Suppose ${\xi} \sim logNormal_d ({\mu},{\Sigma})$, where $\mu$ is a d-dimensional vector (called location vector) and $\Sigma$ is a $d \times d$ symmetric positive definite matrix (called diffusion ...
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3answers
97 views

How is it possible for two random variables to have same distribution function but not same probability for every event?

It is completely out of the world for me to hear that such a case exists. I was shocked and could not develop any intuition as to how it is possible. It also breaks my understanding (intuitive) of the ...
2
votes
0answers
17 views

EM algorithm with constrained equation

I am reading a paper where author uses EM for the following equation to find the parameters $\theta$(and $\beta$) : $$ J=\sum_m \alpha_{m}\sum_i\sum_j w_{mij}\log\sum_k \theta_{ik}\beta_{mjk} $$ ...
0
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3answers
61 views

Hide and seek game

$A$ and $B$ go to the Senate to play a game of Hide-and-Seek. There are $100$ rooms in the Senate, and $B$ picks one of them and hides there till the game ends. $A$, at the beginning of every turn, ...
0
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0answers
30 views

Perturbed density of eigen-states of a 3 diagonal matrix

How does the density of eigen-states ($D(\lambda)$ is defined as $D(\lambda) d\lambda$ = Number of states in the range $\lambda ... \lambda + d\lambda$) of the following tridiagonal matrix ($A$) ...
0
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0answers
18 views

Normal approximation to Poisson-binomial and beta-binomial distributions

I am looking for normal approximations (preferably without any extra expansion terms) for Poisson-binomial and beta-binomial distributions. In particular, something better than the standard normal ...
0
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3answers
35 views

Bivariate Random Variable

Let $(X, Y)$ be a bivariate random variable with support $$S = \{ (x, y) \mid 0 < x < 7, x < y < x + 2 \}$$ and its joint pdf $f(x, y) = 1/14$ for $(x, y) \in S$. (A) Find the ...
1
vote
1answer
43 views

Dirichlet distribution when parameters $\rightarrow\infty$

I am reading a paper where they model a $\overrightarrow{\pi}$ random vector which is Dirichlet distributed in this way: $$ \overrightarrow{\pi}|\alpha\overrightarrow{w}\sim Dir(\alpha w_{1}+1, ...
4
votes
2answers
35 views

Sampling distribution of $Y = \frac{\ln U_1}{\ln U_1 + \ln (1 - U_2)}$, where $U_i \sim U(0,1), \forall i$

For this problem I have used the fact, $-2 \ln U \sim \chi^2_{(2)}$. But I have doubt on the independence of numerator and the denominator which are $\ln U_1$ and $\ln U_1 + \ln (1 - U_2)$. If they ...
0
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0answers
35 views

$\chi^2$ distribution Stoch. increasing in non-centrality parameter

i.e for fixed $\nu>0$ if we have $\gamma_2 > \gamma_1>0$ then $\chi^{2}_{\nu}(\gamma_2)\succeq\chi^2_{\nu}(\gamma_1)$ where '$\succeq$' denotes stochastically larger. The convention that I ...
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0answers
23 views

Distribution of Difference of Ordered Values Drawn From A Normal Distribution

This question has come up at least twice now when I was trying to estimate something*. I could always write out the integral or find it computationally but I'm hoping someone will give me an exact ...
0
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0answers
19 views

Bounds on Chi Squared Distribution

Consider the following hypothesis test: $\mathbf{X} =(X_1,\cdots, X_k) \sim $Multinomial$(n,\mathbf{p})$ and $H_0 : \mathbf{p} = \mathbf{p}_0 = (p_1,\cdots, p_k)$. I know to test this, we construct ...
1
vote
2answers
33 views

Probability distribution of a linear function of a continuous random variable

If $X$ is a continuous random variable, under what condition does the following relation hold true? $$\mathbb{P}(X < k) = \mathbb{P}(aX+b < ak +b)$$ Is the above relation dependent on the PDF ...
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2answers
49 views

If you flip a quarter, what are the odds that you will land on that little piece on the edge? [duplicate]

If you were to flip a quarter what is the probability you will get the quarter to land on it's little edge? How would you calculate this? Assuming the probability is not $0$ and is instead a really ...
1
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0answers
22 views

Name for a constrained Poisson-like bridge process

I have a sequence $t_i$ for $i=0,2,\cdots,n$ of integer jump times with $t_0=0$ and $t_n=n$ such that the waiting time $t_{i+1}-t_i$ has distribution density $f_i(t)$. So it's kind of like a Poisson ...
1
vote
1answer
31 views

X is uniformly distributed on the interval [1,4] and Y = sqrt(X)

Suppose that X is uniformly distributed on the interval [1,4], and that Y = sqrt(X). Evaluate E(Y) and Var(Y). I know the formulas to get the expected value and variance of a uniform distribution. ...
0
votes
1answer
29 views

Normalized sum of uniformly distributed random variables

Let $X_1,\dots,X_n \sim U([0,1])$ be $n$ i.i.d. random variables uniformly distributed over $[0,1]$. Define for all $i = 1,\dots,n$, $Y_i = X_i/\sum_{i = 1}^n X_i$. Does this correspond to some ...
1
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0answers
22 views

Leibniz rule for probability distribution with infinite support.

Let $f$ be the pdf of a non-negative random variable $X$ with finite moments of all orders, i.e. $E[X^n]<+\infty$ for all $n \in \mathbb N$. May I apply Leibniz's rule and infer that $$\frac{d}{d ...