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

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Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
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4 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
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25 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
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1answer
32 views

Martingales problem how to

I am unsure how to approach the following question. Given $\{X_1,X_2,...\}$ let $\displaystyle S_n=\sum_{i}^n X_i$ and $F_n=\sigma(X_1,...X_n)$. Suppose that for all $n\geq 1$, $\mathbb ...
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24 views

Show that $Y_i$ is independent of $Y_j$ for any $i$ not equal to $j$

Let $\{X_1,X_2,\ldots\}$ be independent, identically distributed, absolutely continuous random variables. Let $Y_n=I\{X_n>\max(1< i < n)\}$ for $n=2,3,\ldots$ a) Show that $Y_i$ is ...
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11 views

link between standard normal CDF and normal CDF composed with square root

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$ Let $F_{Y_n}$ ...
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1answer
18 views

Formula needed for calculating probability of recurring events

I'd like to find an answer for calculating the following recurring events: You have X opportunities of picking a ball from a sack. Every time after a ball is picked, the ball is returned to the sack. ...
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1answer
23 views

Average number of rolls before going broke

I have a difficult probability question to resolve. Say you have 2 chances to roll a dice. If you roll a 6, you're awarded 2 additional rolls. You can receive infinite number of additional 2 rolls ...
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10 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
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1answer
22 views

$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise. I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt ...
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1answer
11 views

Expected value, variance and probability from a joint distribution function

Lets say I am given the following table that shows the joint probability function of X and Y: $$\begin{array} \\{}&y=1&y=2&y=3 \\x_=1&0.1&0.2&0.1 ...
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1answer
11 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
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1answer
35 views

Roll Dice- Expected Winnings [on hold]

I have a problem like this: At a charity game you pay \$1 to roll a die. If you roll a 6, you get \$5. Otherwise, you get nothing. How do I set up a probability distribution and what is the ...
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2answers
21 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
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1answer
21 views

Probability of same last four digits of a telephone number

Hi all this is really my first post here .. Yesterday I was talking to a girl and asked her for her phone number . Once she gave it to me we realized that we got exact same last four digits . Hence ...
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1answer
30 views

Galton Machine and Unpredictability

We are all familiar with the Galton Machine and the images of the balls cascading through the device and ending up in bins which ultimately show a likeness to the binomial distribution. Most everyone ...
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1answer
9 views

Probability of 2 of three independent events occuring

Three objects are thrown at a target. The probabilities the individual objects will connect with the target is .75, .85 and .90. Find the probability that at LEAST two of the objects hit the target? ...
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10 views

Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
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1answer
26 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
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12 views

p.d.f and distribution of multivariate normally distributed variables

Suppose $X\sim N(\mu,V)$ where $\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ $V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ a) ...
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2answers
21 views

Find Normalizing constant

let $f(x,\theta)=C_\theta \exp(-\sqrt{x}/\theta)$ where $x$ and $\theta$ are both positive. Find the normalising constant $C_\theta$. I get $C_\theta=\sqrt{2}/\theta$ but my book says ...
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29 views

Average minimum distance

I posted a question earlier here and someone pointed out that it might not be possible to find a closed form solution due to the elements of $\mathbf{g}$ and $\mathbf{f}$ defined below coming from a ...
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7 views

Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
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1answer
72 views

Challenging question about probability [on hold]

A manufacturer of plastics claims that its waste is managed in such a way that benzene, a harmful chemical, cannot get into the local ground water. People living near the factory are not so sure. A ...
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3answers
60 views

Finding probability of a random point [on hold]

Consider a square with sides of length 1 and the bottom left corner at (0;0). Choose a point P randomly within the square. Show that the probability that P is closer to (0;0) than to (0.5, 0.5) is ...
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1answer
14 views

Distribution of a function of a uniform random variable.

I ran across this example the other day and was surprised at how stumped I was. Suppose $U$ is a uniform random variable on the interval $[0,1]$. Let $F = \frac{1}{U+3}$. What is: ...
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1answer
18 views

Joint distribution and Integration

I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim: Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and ...
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29 views

Average minimum distance between two random vectors

Let $\mathbf{y_1} =\begin{bmatrix}g_1x_1 & g_2x_1 & \dots & g_Nx_1 \end{bmatrix}$ and $\mathbf{y_2} = \begin{bmatrix} f_1x_2 & f_2x_2 & \dots & f_Nx_2\end{bmatrix}$. All the ...
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15 views

Queueing and probabilities

Messages are transmitted from low speed terminals and arrive at a message concentrator at a Poisson rate of $600$/hour. They are held in a buffer until a hi-speed trunk line is free to transmit them. ...
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8 views

Projected multivariate normal distribution

Assuming a set of points (x_i,y_i,z_i) is distributed according to a multivariate normal distribution with a given mean and an isotropic covariance matrix, i.e. ...
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2answers
30 views

Probability distribution morphing from Gaussian to heavy tail

I require a probability distribution which morphs from a something similar to a Gaussian (image 1) to something with a heavy tail on one side (image 2) based on some parameter. Can someone give me a ...
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17 views

Formula for a Skewed Distribution

I'm trying to create a model of some data, using a skewed normal distribution. I have the following data: Mean Median Standard Deviation from the mean Standard Deviation from the median I've been ...
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27 views

CDF of the random variable $U$ defined by $U=Y^n$

I was working on a practice exam today and I got stumped on the following question: The continuous random variable $Y$ has probability density function $f$ given by $$f(y)=\cos(y),\quad 0\le ...
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2answers
21 views

Probability of 2 independent events and 1 mutually exclusive

Let $\Omega$ be the sample space for an experiment $E$ and let $A,B,C\subset \Omega$. If events $A,B$ are independent, events $A,C$ are disjoint, and events $B,C$ are independent, find $\Pr(B)$ if ...
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1answer
20 views

Expected value of probability distribution [on hold]

A random variable $X$ is distributed in $[0, 1]$. Mr. Fox believes that $X$ follows a distribution with cumulative density function (cdf) $F$ : $[0; 1] \rightarrow [0; 1]$ and Mr. Goat believes that ...
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20 views

Probability: extreme value theory

I would need some help for a project, thank you so much in advance! Let $(X_i+n)/2 \sim \mathrm{Binomial}(n,1/2)$ for all $i$ integer such as $1\le i\le2^n$ Let $X_d=\max_{1 \leqslant i \leqslant ...
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3answers
38 views

Marginal density function of a new variable

Question: Let $X \sim U[0,1]$ and $Y \sim U[0,1]$ be independent random variables. By considering the random variables $U=Y$, $V=XY^{2}$, or otherwise, find the probability density function of $V$. ...
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15 views

Integrating a half-normal pdf [migrated]

The pdf of a half normal distribution is: $\frac{\sqrt{2}}{\sigma\sqrt{\pi}}$exp$\left(-\frac{x^2}{2\sigma^2}\right)$, $x>0$ ...
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1answer
39 views

Inferring symmetry of a distribution from its marginals

Let $X=[X_1,\ldots,X_n]$ be a continuous random vector of size $n$ with density function $f_X(x_1,\ldots,x_n)$. If all the marginals \begin{align*} \int \ldots \int f_X(x_1,\ldots,x_n)\, ...
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1answer
31 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
27 views

Die Question with Random Variables

I have a homework question: Roll a fair die, and let d be contained in $\{1,2,3,4,5,6\}$ . Then sample $d$ independent uniform random variables on $[0,1]$ and let $Y$ be the maximal of these random ...
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19 views

Distribution and Laplace transform

I'm having trouble understanding this problem from Resnick's Adventures in Stochastic Processes: The problem says: Suppose $F$ is a distribution of a positive random variable and $p_k \geq 0, ...
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2answers
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Conditioning on a joint exponential distribution

I'm working on a problem from Hogg (7.3.4) where there is a joint pdf. $$f(x,y) = \frac{2}{\theta^2}e^\frac{-(x+y)}{\theta}$$ Valid for $ 0 < x < y < \infty$ As part of the problem, I need ...
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9 views

$\chi^2$ P values comes to be zero

I want to find the P Value of $\chi^2$ (Pearson), in order to see if there is a significant difference between the given two following distributions: ...
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1answer
26 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
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14 views

approximating a probability density

Let $f(x)$ be the probability density of a random variable $X$. Let the support of $f(x)$ be positive reals. If $f(x)$ is sufficiently smooth then one can approximate it with its Taylor series cut off ...
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1answer
15 views

Moment generating function of $X \sim N(Y,1)$ [on hold]

How can I find the moment generating funcion of a random variable X such that $$X \sim N(Y,1)$$ where Y is a random variable with distribution $Y \sim exp(1)$ and $N$ represents the normal ...
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0answers
16 views

Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
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92 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...