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

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0
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3answers
48 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
votes
1answer
47 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
60 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
votes
1answer
56 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.: $$\frac{\partial f}{\partial t}+v\cdot\nabla_{x}f=0$$...
4
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1answer
107 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 continuous for each $t$. Note that ch.f. means "...
0
votes
1answer
31 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
42 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 $i$. $S_{30}=X_1+...X_{30}$....
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2answers
51 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 $.3$. ...
1
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1answer
57 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 $f_X,_Y(...
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votes
2answers
69 views

Independence, conditioning, and correlations

Suppose $X$ and $Y$ are independent random variables uniformly distributed on $[0,1]$. Suppose we consider a conditional distribution of $X$ and $Y$ on some event $C$. Is it possible that these ...
1
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0answers
19 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
35 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
39 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
vote
1answer
94 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
vote
1answer
59 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 $...
1
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0answers
49 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 ...
1
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0answers
304 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
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1answer
95 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
votes
1answer
67 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
votes
0answers
49 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 \mathcal{N}(\...
0
votes
2answers
68 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 \...
1
vote
1answer
52 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 $f_{X,Z}(x,z)=\frac{1}{2\pi\sigma}...
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1answer
61 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 ...
1
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1answer
26 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
votes
1answer
159 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
137 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
69 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 ...
1
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1answer
77 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
171 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, ...
1
vote
0answers
32 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 ...
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votes
1answer
40 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 $M_x(...
1
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2answers
65 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 ...
0
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0answers
57 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
40 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
31 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 ...
0
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0answers
22 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?
1
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1answer
59 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 $\...
1
vote
1answer
77 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
34 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 that $...
1
vote
1answer
30 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
40 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
43 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
38 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 ...
1
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0answers
32 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
49 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 ...
0
votes
0answers
54 views

Geometric Mean of Random Variables

I measure a series of $n$ objects [O_1, O_2, O_3, ..., O_n]. Because those measurements are quite hard to perform, I have quite a lot of measurement error and ...
1
vote
0answers
128 views

Product of Wishart and inverse Wishart distributions

Let $$ X \sim \mathcal{W}_{q} (n, \Sigma) \; \; n > q$$ and $$ Y \sim \mathcal{W}^{-1}_{q} (n, \Sigma^{-1}) \; \; n>q$$ Where $\mathcal{W}$ denotes the Wishart distribution and $\mathcal{W}^{-1}...
0
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1answer
54 views

Coutinuous distribution in Probability

If suppose there is an interval $[a,b]$ then choosing a number from it is equal probable and a number can be any real number within the interval. Is it a case of continuous distribution ? How to ...
2
votes
3answers
72 views

Prove that $f(x)=exp(-x-e^{-x})$ for $x\in \mathbb{R}$ is a p.d.f and find the c.d.f.

Prove that $f(x)=exp(-x-e^{-x})$ for $x\in \mathbb{R}$ is a probability density function and find the cumulative density function. I think that by proving that $f(x)$ is a pdf, it should be fairly ...
0
votes
1answer
124 views

Jar and Ball Probability Distribution

If I have 8 jars, each jar contains 5 unique ball types. However, I know that I have 20 unique ball types out there. So, I have balls labelled from B1, B2, B3, ...B20 to put into 5 jars. Let's say ...