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

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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 ...
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1answer
1k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
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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 ...
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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 ...
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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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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 $i$. ...
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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 ...
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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 ...
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1answer
2k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
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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 ...
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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 ...
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1answer
19 views

The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
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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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1answer
70 views

E(XY) = E(X).E(Y|X) . Is this true for mean = zero.

I know that Joint Probability density function for two random functions $X$ and $Y$ $$P(XY) = P(X)\cdot P(Y|X)\tag{1}$$ But I just read in a set of lecture notes that for E(X)=E(Y)=0 $$E(XY) = ...
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2answers
305 views

Variance of a piecewise pdf

How can I calculate the variance of a piecewise continuous function? For example, $f(x)=0.2$, $0 \le x \le 0.5$ ; $2.4x-1$, $0.5 \le x \le 1$
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0answers
70 views

Sums of Power Law random variables

Suppose $F$ is a Pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , \dots, X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
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1answer
498 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...
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1answer
407 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
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1answer
54 views

Find distribution function and pdf for a random variable based on a uniform distribution

Let $X$ be uniform on $(0,1) $. Definition of $Y$ is as following: $$ Y=\begin{cases}e^X \quad &(0\le X<0.5) \\ \log X\quad &(0.5\le X\le 1)\end{cases}$$ find Definition function ...
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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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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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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 ...
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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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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 = ...
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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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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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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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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? ...
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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 ...
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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)- ...
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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 ...
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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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2answers
537 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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\} ...
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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
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
19 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
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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 ...
2
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1answer
54 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
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3answers
18k views

How can a probability density be greater than one and integrate to one

Wikipedia says: The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. and it also says. Unlike a probability, a probability ...
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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 ...
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2answers
78 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: \begin{equation} p(x)=\int ...
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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 $\mu$ ...