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

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2
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0answers
29 views

Gaussian Distribution Under Orthogonal Transformation

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. ...
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1answer
16 views

Find the distribution of some random variable connected to Wiener Process. Please, check my solution.

I need to find a distribution of $ 5W_1-W_3+W_7 $, where $W_t$ stands for Wiener Process $W_t\sim\mathcal{N}(0,t)$. Is this solution right? $E(5W_1-W_3+W_7)=5E(W_1)-E(W_3)+E(W_7)=0$ and since ...
2
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1answer
48 views

Will someone please explain multivariate normal distributions with a real-life example?

I understand a concept best when I see it being applied in the real world.
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3answers
33 views

Bernouli Trial Probability of Stopping After X Trials

The probability of a trial being a success if 0.30 Trials are repeated until 6 are successful. I'm asked to find the probability that the trials are ended after the 7th. (The 6 successful trials ...
0
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3answers
61 views

Probability of not choosing from a lot [closed]

Say I have a lot of $1000$ televisions. $10$ out of every $1000$ are defective. When I choose $30$ televisions to test them, how would I calculate the chances of not choosing a defective one from ...
0
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1answer
32 views

Find the $p_{Y|X}(y|x)$ without the jointly probability

Let the distribution $Y = X + N$. Where $X$ and $N$ are independents and they have distinct distributions. I have $f_X(x)$ but I don't have the $f_{XY}(x,y)$ to use, for example, the following ...
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0answers
50 views

Determining the Expected value of a random variable

Suppose we have a Poisson process of parameter $\lambda$. Each event of this Poisson process represents a start date of a period which duration is a random variable that follows an exponential ...
0
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1answer
30 views

Conservation of Kinetic Energy in Vlasov-Poisson System

I'm studying the very basics of kinetic theory in Vlasov Poisson Systems, and the first equation I'm studying is the free transport equation, i.e.: $$\frac{\partial f}{\partial ...
1
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0answers
15 views

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is coutinuous for each $t$. Note that ch.f. means ...
0
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1answer
14 views

Finding a CDF given a PDF using summations

I am in a prob and stats class and we have just begun our discussion on discrete random variables. I am given a pdf of $$ f(x) = \left\{\begin{aligned} &x/10 &&: x = 1,2,\ldots,4\\ ...
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0answers
34 views

Poisson distribution of a sum.

Suppose the number of robberies of a clothing store in a random day is a random variable with Poisson distribution with $\lambda=5$. $X_i$ is the number of robberies in day $i$. ...
0
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1answer
29 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 ...
1
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1answer
28 views

Determine the expectation E(XY) of Joint PDF

I am practising some exam questions and am failing to understand the problem at hand. I believe I am supposed to take the double integral of the joint PDF that can be calculated by noting that ...
1
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0answers
16 views

Joining heterogeneous, discrete probability mass functions

Suppose we have a collection of discrete probability mass functions with different ranges, all of which are from 0 to some positive integer. As a simple example, we might be rolling 3 6-sided dice, 1 ...
1
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1answer
31 views

What's the probability of obtaining exactly 3 C's out of 10 exams?

The result of an exam consists in three possible grades: A, B and C, each with equal probabilities. What's the probability of obtaining exactly three C's out of 10 exams? And what's the probability ...
2
votes
1answer
27 views

Do the set of all standardized moments of a dataset completely and uniquely define it?

I have two datasets, 'A' and 'B', comprising N measurements of one quantity, that I would like to compare to the results of a simulation, let's call this last dataset 'S'. This comparison got me ...
1
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1answer
93 views

A question in combinatorics

Given a sequence of $0$s and $1$s think of it as blocks of $0$s and $1$s. Like $0001101001$ is a sequence of blocks $000$,$11$,$0$,$1$,$00$,$1$ How may ways can one pick $t$ bits from a $0/1$ ...
1
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1answer
55 views

Is this PMF or PDF?

I am reading a technical report on expectation-maximization (EM) algorithm (http://melodi.ee.washington.edu/people/bilmes/mypapers/em.pdf) and I am confused about something. For HMMs, it defines ...
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0answers
29 views

Finding a likelihood function given binary observed data

I'm having trouble really understanding the terms used for this homework question, and what I am actually supposed to be doing, given the actual data for the problem. Below is the problem: Suppose we ...
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0answers
20 views

How to calculate Fisher Information (FI) matrix for Multivariate Normal Distribution (MN)

Below is the gradient (score) of the MN log likelihood function L for n=1 observation. I originally attempted to calculate the Hessian matrix but ran into difficulty calculating 2nd order derivatives ...
2
votes
1answer
43 views

Wasserstein distance between distribution functions

It is well-known fact that if we have two DFs F and G with finite second moments, then one can calculate the Wasserstein distance between them using this formula: $$ W_2^2(F,G) = \inf E(ξ-η)^2 = ...
3
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1answer
24 views

What is the probability that two univariate Gaussian random variables are equal?

Let $X_1$ and $X_2$ be two independent univariate Gaussian random variables, s.t. $$X_1\sim \mathcal N (m_1,\sigma_1^2)$$ $$X_2\sim \mathcal N (m_2,\sigma_2^2)$$ So now what is $P(X_1=X_2)$? I tried ...
2
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0answers
28 views

Distribution of sum of absolute values of 2D Gaussian

It was a while back I read probability theory and I've stumbled on a question in my work I'm not to sure about. I have a position a=(x,y)+g with a added 2D Gaussian noise g $\in ...
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2answers
53 views

How to compute probability related to a difference of two random variables

I am studying Joint Probability Distributions and Random Samples. I have a function for a probability distribution, defined as: $ f(x, y) = K(x^2 + y^2)~~~~~~~~~ 20 \leq x \leq 30, ~~~20 \leq y ...
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1answer
34 views

Conditional density, bivariate normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independent. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...
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1answer
38 views

Calculating probability distribution under given constraints

I recently asked a question about the construction of a random variable under given constraints (see: Construct a random variable under given constraints). The only answer to my question suggested a ...
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votes
0answers
28 views

Probability density standard normal distribution

Me and my friends from math class don't really know how to start with the following question. No matter how long we look at it, we can't get any ideas to solve this. Is there somebody who can help us? ...
1
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1answer
16 views

How to Solve Multiple Stopping Problem with a Known Payoff Distribution

I'm interested in learning how to optimally solve a multiple stopping problem with a known payoff distribution, like the following: You are observing a sequence of forty $(40)$ opportunities, each ...
0
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1answer
40 views

Compound of Exponential and Inverse Gamma Parameter

I am trying the prove the following: Show that an exponential random variable such that the inverse of the parameter is gamma-distributed is Pareto-distributed. More precisely, show that if $$X | M = ...
1
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1answer
25 views

Application of Compound Poisson Process

I am trying to solve the following application problem: The life T (hours) of the lightbulb in an overhead projector follows an Exp(10)-distribution. During a normal week it is used a Po(12)- ...
0
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1answer
28 views

Recover the distribution of a Binomial random variable from its Characteristic Function

Hoping someone could show how to use the Characteristic Function of a binomial r.v. to recover its distribution. Using the inversion formula to recover the pdf of a r.v. with a continuous ...
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1answer
56 views

Why is the strong law of large number stronger than weak law? [closed]

The weak law is easy to prove, but the strong law (which of course implies the weak law, by Egoroff’s theorem) is more subtle. I'd like to know for which mathematical reason is the strong law ...
-1
votes
1answer
33 views

How to get uniform distribution with two dice rolls?

The sum of two dice rolls will not have uniform distribution. Never realized... Is there an easy way to cheat? Will this work? 1st die roll, 1-6... 2nd die roll, if 1-3, add 0 to first die, if 4-6, ...
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0answers
24 views

What function describes the frequency for each unique ratio for all possible expansions n over d where n<d?

I am hoping to solve the following problem for a scientific investigation, which relies on the probabilites of all possible expansions. What function $f(r)$ describes the frequency for each ratio for ...
0
votes
1answer
25 views

Distribution from moment generating function [closed]

Moment generating function for $ X ~ (\vec{\mu}, \Sigma) $ is of form $ M_x(t) = exp( t^T\vec{\mu}+\frac{1}{2}t^T\Sigma t)$ The random variable $X = [T_1, T_2]^T$ has moment generating function ...
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2answers
49 views

Construct a random variable under given constraints

In preparation for a probability examination, I am working on the following problem. Problem A box contains three white balls and ten black balls. Balls are drawn without replacement until all the ...
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0answers
23 views

Improper integral over product of exponentials: $\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$

I'm looking for a way to evaluate following integral $$ \int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx $$ f(x) resembles however a complex simulation and can therefore ...
0
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0answers
20 views

Exlain the significance of the uniform random variable for the simulation of random variables

I can think of the "Universality of the Uniform": Given an Unif(0,1) r.v., we can construct an r.v. with any cts distribution we want. Conversely, given an r.v. with an arbitrary cts ...
0
votes
1answer
24 views

Probability of Playing Darts

We have a dartboard with radius $1$, the dart will always hit the dartboard. The hitting point of the dart is uniformly distributed, with a stochastic vector $(X,Y)$. Now I want to determine the ...
2
votes
0answers
18 views

Law of a supremum of random variables

Let $(B_t)_{t\geq 0}$ the standard brownian motion (with $B_0=0$), $p$ be a real number greater than $1$ and $q$ its conjugate number. Prove that $X_p=\sup _{t\geq 0}(|B_t|-t^{p/2})$ is a.s. strictly ...
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0answers
13 views

# of Crossing of pairs continuous distribution functions and # of crossing of their inverse

Suppose $F_X$ and $F_Y$ are two continuous probability distributions that cross only twice. Does that imply that $F_X^{-1}$ and $F_Y^{-1}$ also only cross twice?
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1answer
32 views

Distribution of arcsin of a uniform random variable

Question: Find the law of $\arcsin(X)$ where $X\sim Unif[0,1]$ and where $X\sim Unif[-1,1]$ My attempt: We say $f_X(x)=Unif[0,1]$, and that $Y=\arcsin(X)$ We say $x=\phi^{-1}(y)=\sin(y)$ and have ...
0
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0answers
47 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
2
votes
1answer
23 views

Joint density of normal random variables

Let $Z=X+Y$ where $X$~$N(\mu,\sigma^2)$ and $Y$~$N(0,1)$ are independents. Find the joint density of Z and X. This is the first time I see something like that, look what I did below: I know ...
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1answer
20 views

Question about uncorrelatedness of random variables and distributions

I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa? Also, if I know that a joint distribution of two ...
0
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0answers
32 views

Find the distribution of $Z=\frac{X_1+X_2}{X_1X_2}$, where $X_1$, $X_2$ follow normal distribution

Lets assume $X_1$, $X_2$ follow normal distribution. I am looking for the distribution of: $$Z = \frac{(X_1+X_2)}{X_1*X_2} $$ This is what I have thought so far: The distribution of the ...
1
vote
1answer
31 views

Expectation of the time difference between starting times in queueing theory

Consider 2 independent, parallel $M/M/1$ queues $Q_1, Q_2$ with identical arrival rate $\lambda$ (corresponding to an exponential random variable $A \sim \text{Exp}(\lambda)$) and service rate $\mu$ ...
2
votes
0answers
22 views

The probability that two or more successive tasks with Weibull distributed lengths have completed?

I have a set of independent tasks whose lifespan/time it takes to complete seems to fit nicely into a Weibull distribution. The tasks are to be handled one by one, sequentially. As far as I ...
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0answers
24 views

Suppose that $U$ is uniformly distributed on $[0,1]$. Given its p.d.f. and c.d.f, find $P(U<a|U<b)$ for $0<a<b<1$.

Suppose that $U$ is uniformly distributed on $[0,1]$. Find $P(U<a|U<b)$ for $0<a<b<1$. We know that the p.d.f. of $U[a;b]$ is $f_X(x)=\begin{cases}\frac{1}{b-a} & :\text{for }a ...
1
vote
2answers
30 views

Sufficient parameters for a probability distribution

We know that a Gaussian distribution can be constructed if its first two moments i.e. its mean and covariance are known. Is there any other standard distribution whose construction requires the ...