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

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91 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
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1answer
503 views

A basic difference between moment generating function and Laplace transform

I have read in a probability book that the advantage of dealing with Laplace transform (for non-negative random variable) rather than moment generating function is that Laplace transform is always ...
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1answer
180 views

Distribution of the fractional part of a sum of two independent random variables

Let $X$ and $Y$ be independent random variables. $X$ is a uniform random variable on $[0,1]$. Let $Z=X+Y-\lfloor X+Y\rfloor$. What is the distribution of $Z$? Let $X_1, X_2, \dots, X_n$ be IID with ...
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2answers
151 views

What is the probability of a multidimensional rectangle?

Assume given a probability measure $P$ on $(\mathbb{R}^p,\mathcal{B}_p)$, where $\mathcal{B}_p$ denotes the $p$-dimensional Borel-$\sigma$-algebra. Let $F$ denote the $p$-dimensional CDF for $P$, ...
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1answer
574 views

Relation chi-square and student-t distribution

First I want to prove that the sum $Y_1+...+Y_n$ where $Y_i=X_i^2$ and $X$ is standard normally distributed has density $f_n(x)=c_n x^{n/2-1}e^{-x/2}1_{x>0}$ I do not want to derive it, I would ...
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1answer
145 views

Correlation Coefficient - $\rho(X,Y)$.

If I have two aleatory variables $$X=\begin{pmatrix}2&3&4&5&6&7&8&9&10&11&12\\ ...
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2answers
52 views

How to show this equality of probability on the unit disk

I came up with this problem which I think is so intuitive but fails to give more rigorous and convincing argument. Let $(X,Y)$ be uniformly distributed in the disk $D:=\{(x,y):x^2+y^2\le 1\}$ For ...
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1answer
195 views

Why use infimum in definition of Quantile function

I looked up the definition of Quantile function in Wikipedia, it is said that: The Quantile function is $Q(p)=\inf\{x\in R:p\le F(x)\}$ for $F:R\to(0,1)$, for a probability $0<p<1$, the ...
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2answers
1k views

Calculate density and cumulative distribution of: Y = X1 - X2

Hi I have this question in my book (preparing for the exam) and I can't seem to find the answer.. X1 and X2 are independent and identically distributed. They are continuous and uniform over [-1,1]. ...
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2answers
121 views

Homework Help. Finding the PDF of a new random variable.

$ f_{XY}(xy) = \begin{cases} 2e^{-x - 2y}, & \text{if $x>0$ and $y>0$} \\ 0, & \text{otherwise} \\ \end{cases} $ a new random variable Z is defined as $Z = X + Y$. Find the ...
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3answers
96 views

Find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$.

If $X_1,X_2\sim \text{Normal} (0,1)$, then find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$. I solve problems where transformation is given and I need to find the distribution. But here I ...
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1answer
26 views

Give example of a distribution.

Give examples of distribution (1) such that $X$ and $1-X$ have the same distribution. (2) such that $X$ and $\dfrac1X$ have the same distribution. For the first one I think $X$ is ...
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0answers
186 views

A basic probability question on uniform distribution

Suppose that independent trials, each of which is equally likely to have any of $m$ possible outcomes, are performed until the same outcome occurs $k$ consecutive times. If $N$ denotes the number of ...
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1answer
90 views

Phase-type distribution - moments

I was doing my best to calculate moments of phase-type distribution. Density of phase-type distribution is $$f(x)=\alpha e^{Sx}S_{0}$$ ($\alpha$ is $1\times m$ vector; $S_{0}$ is $m\times 1$ vector; ...
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0answers
98 views

Discrete probability distribution for a multi-stage experiment

Say we choose $n$ experiments to perform (with replacement) from the experiment types $1\dots C$, where an experiment of type $i$ is chosen with probability $q_i$. Experiments of type $i$ have ...
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0answers
47 views

Compute the joint probability of two “ordering” events

Let $\{o_1, o_2, o_3, o_4\}$ be a set of objects, each one associated with a score $s(o_i)$. This score is uncertain, and thus described by means of a probability density function $f_i$. When a total ...
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0answers
49 views

Expectation values in directional statistics

In the context of Variational Bayesian Inference I am facing the following problem: Let $\alpha$ follow a "von Mises" distribution with mean $\mu$ and concentration $\kappa$. Does there exist a ...
2
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0answers
269 views

Example of Google page ranking algorithm?

I read about the Google page ranking algorithm from here http://en.wikipedia.org/wiki/PageRank . My question is why only outbound links are used in page rank calculation? Do inbound links not ...
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2answers
79 views

Probability of rolling a $6$ between $90$ and $110$ times when rolling a die $600$ times

Show that if you roll a fair die $600$ times, then with probability at least $1/6$ you will get the roll $6$ between $90$ and $110$ times. This is what I have so far Let $E(x) = 100 \Rightarrow ...
2
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1answer
5k views

Probability Distribution of Rolling Multiple Dice

What is the function for the probability distrabution of rolling multiple (3+) dice. The function is a bell curve but I can't find the actual function for the situation. Example, what is the function ...
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1answer
357 views

Joint distribution of two uniform RV's

Let $X \sim U(0,\theta_1) \text{ and } Y \sim U(0,\theta_2)$ be independent RV's. We're interested in $\mathbb{P}[X-Y\le z]$. I've done the convolution (or double integrals), denote $Z=X-Y$ ...
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1answer
46 views

A definite integration problem on law of large number

The problem is given as follows: If $g(x),h(x)$ are continuous function on $[0,1]$, satisfying $0\le g(x) <M h(x)$, where $M$ is a nonzero constant. Prove that $$\lim_{n\to \infty} ...
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0answers
63 views

Distribution of $pX+(1-p) Y$

We have two independent, normally distributed RV's: $$X \sim N(\mu_1,\sigma^2_1), \quad Y \sim N(\mu_2,\sigma^2_2)$$ and we're interested in the distribution of $pX+(1-p) Y, \space p \in (0,1)$. ...
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1answer
40 views

About the connection of $L^2$-convergence and convergence in distribution.

Let $(T_{1,n})_{n\in\mathbb{N}}$ and $(T_{2,n})_{n\in\mathbb{N}}$ two sequences of real valued random variables in $L^2(\Omega,\mathbb{A},P)$. Suppose that ...
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1answer
255 views

Question on M/M/s queue

costumers arrive to a service station according to a poisson prossees and on average 2 during an hour.the service times and independent of the arrivals and internally independent with mean 45 minuts ...
3
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1answer
62 views

Implied prior with relationship $y=\text{arccot}(x)$

I'm trying to solve an exercise, which I think I have almost managed to solve but not quite. Any help would be appreciated! So, what we have is a vector which we obtain by norming the vector ...
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2answers
41 views

Probability example for homework

There's a highway between two towns. To reach the other, people must pay 10 dollars for cars and 25 dollars for bigger vehicles. How big income can we expect if the 60 percent of vehicles are cars and ...
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2answers
395 views

Angular distribution of lines passing through two squares.

Let's say I've got two squares with side length $d$ that are held parallel at a distance $m$ apart. Suppose that particles are randomly falling from above in such a way that the polar angle ...
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1answer
445 views

Find the Posterior distribution- prior: $exp(1)$, likelihood: $poisson(\lambda)($

I have a prior $\lambda \sim exp(1)$ and a likelihood $X \sim poisson(\lambda)$, and I observed in a sample of $n=5$ a mean of $3$. What is the posterior distribution of $\lambda$? Here is my ...
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2answers
32 views

how do determine the distribution of outcomes for a given probability?

For a game I generate various block types given certain odds. Say, there's a $0.001$ chance for the karma block. If a typical game has $600$ blocks, what's the distribution of games that have $0$ ...
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0answers
28 views

How to simulate a sequence of partial sums $(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),$ given some properties.

I want to generate/simulate a sequence of partial sums. $$(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),\text{ for }1 \leq n \leq 100$$ Let $W$ be a random variable such that: $W \thicksim ...
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2answers
46 views

First moment inequality implies tail distribution inequality?

Let $U,V$ be two continuous random variables, both with continuous CDF. Suppose that $\mathbb E V \geq \mathbb E U$. Can one conclude that $\mathbb P(V> x) \geq \mathbb P(U>x)$ for all $x\geq ...
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1answer
35 views

$X$ random uniform variable, what is the cumulative distribution of $|X|$?

Let $X$ be a random uniform distribution on $[-2,1]$. What is the cumulative distribution of $|X|$, i.e. $G(x) = p(|X| \leq x)$ ? The cumulative distribution of $X$ is $F(x) = \dfrac{x+2}{3} $. ...
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1answer
126 views

Random variable with density proportional to a function and finite in some points

Let $X$ be a random variable on $[-1,3]$ with density $f(x) = k x^2$ (with $k \in \mathbb{R} $ to be determined) on $[-1,3]$ apart from some points s.t. $p(X=-1) = p(X=3) = \dfrac{1}{4} $ and $p(X=0) ...
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1answer
65 views

What does this hint mean and how is it useful to solve the problem?

I am doing a problem on convergence of random variable. There was a hint given, but I am still struggling to understand the hint. Here is the problem: Let $Y_n$ be uniformly distributed on ...
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1answer
84 views

Finding the value of distribution function of a converging random variable

There is this example in a note that I think this is supposed to be a simple problem, but I still find it not as straightforward. Consider a sequence of random variables $X_n\equiv1/n,X\equiv0$. Then ...
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4answers
582 views

exponential distribution - get some meaning

I read that a continuous random variable having an exponential distribution can be used to model inter-event arrival times in a Poisson process. Examples included the times when asteroids hit the ...
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2answers
181 views

A simple yet hard task for (theoretically) Poisson distribution

Sorry if I don't use the words properly, I haven't learnt these things in English, only some of the words. Anyway, I'm practicing to one of my exams and sadly this task seemed more challanging for me ...
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1answer
96 views

correlation of product with its normally distributed factors

If x and y are normally dist. with standard deviation of 10%, and they are independent, then their product X.Y is 71% correlated with Y (or X). I can show this empirically, but how to I prove it in ...
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2answers
390 views

Uniform distribution joint $\to$ marginal

Let vector $(X,Y)$ have a uniform distribution on the set $N = \{ (x,y): x<1,y<1,1<x+y\}$. Determine distribution $X-Y$. So far I've thought of this: \begin{align} P[X | Y=y] &\sim ...
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1answer
41 views

help me with this regarding hypothesis using chi square distribution

The rope used in a lift produced by a certain manufacturer is known to have a mean tensile breaking strength of 1700 kg and standard deviation 10.5kg. A new component is added to the material which ...
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1answer
33 views

this is regarding exponentials distribution

In an office building, the lift breaks down randomly at a mean rate of 3 times per week. The random variable X represents the time in days between successive lift breakdowns. (i) Calculate the ...
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1answer
36 views

What is the purpose to define different moments on a distribution?

What is the purpose to define different moments on a distribution? The first moment is the expectation value of a function, what about the other?
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1answer
334 views

Acceptance sampling schemes for binomial distribution

Two acceptance sampling schemes, A and B, are proposed for deciding whether or not to accept a large batch of items from a production process in which 5% of the items produced are defective. Scheme A: ...
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1answer
60 views

Basic question on the transformation of Exponential distribution.

Why central moments coincide for random variables $V\sim E(a,h)$ and $Y\sim E(h)$ where a=location parameter h= scale parameter.
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3answers
227 views

A question regarding the Poisson distribution

The number of chocolate chips in a biscuit follows a Poisson distribution with and average of $5$ chocolate chips per biscuit. Assume that the numbers of chocolate chips in different biscuits are ...
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2answers
50 views

Probability of two variables of having the same value

Let $X$ and $Y$ be two random variables, whose PDFs $f_X$ and $f_Y$ are uniform. $f_X$ and $f_Y$ may overlap. For instance, they could represent two score distributions for two tuples $x$ and $y$ in a ...
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2answers
139 views

Correct application of birthday problem

The problem is as follows: There are 10 shooters at a shooting range. Each shooter is given 5 bullets. They all begin shooting at 9am and end shooting at 10am, They each shoot all 5 of their bullets ...
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2answers
168 views

Probability Joint PDF

Every night Joe goes to the casino and takes with him an amount of money in dollars, X, that is distributed according to the pdf: f(x) = Ax^2 for 0 < x < 10 where A is a constant that you need ...
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0answers
38 views

Calculating the probabilities of different lengths of repetitions of numbers of length 6

This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4 except now I'm having problem with numbers of length 6. ...