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

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1answer
243 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
429 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
45 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
124 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 ...
3
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4answers
569 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
172 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
383 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
318 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
225 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
43 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
138 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
166 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. ...
0
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1answer
270 views

A quick Poisson distribution problem

A plane drops 535 bombs over 576 fields. How many fields were hit at least twice? I have a feeling this is related to the Poisson distribution, but I'm having trouble identifying the variable which ...
3
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0answers
116 views

Problems sampling from a $pdf$ over $SO\left(3\right)$

I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series: $$ f\left(\omega,\theta,\phi\right)=\sum ...
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2answers
152 views

How do I divide a set of data samples which follow a logarithmic distribution?

I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
0
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2answers
40 views

How to find density function?

$X \sim N(1,4)$ and $Y = 3 - 5X$. How to find the density function of $Y$? I tried first to find the distribution function of Y, but got stuck. $$F(y) = P(Y <= y) = P(3 - 5X <= y) = P(X >= ...
2
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1answer
42 views

Range of the distribution of $(1-X)$ when $X$ follows Beta distribution as $X\sim beta(p,q)$

if $X$ follows beta distribution with parameter $p$ and $q$ where $p>0\quad , q>0$ then $1-X$ follows beta distribution with parameters $q$ and $p$, that is if $X\sim beta(p,q)$ then ...
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1answer
446 views

Probability exponential distribution.

May I please borrow your expertise or could anyone check if I'm on the right track please? Consider customers arriving at a bank. The bank has $2$ types of customers - business and personal. On ...
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2answers
438 views

Closed form for Exponential Conditional Expected Value & Variance

I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution. For example: $$ E(X|x > a) $$ where X is exponential with mean ...
1
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1answer
87 views

Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
0
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1answer
68 views

Boltzmann Distribution With Constraints

I have a problem with showing the existence of Boltzmann distribution given some constraints. Consider $p_1,...,p_n$ a Boltzmann distibution, where $p_i=\frac{\epsilon^{-\beta \cdot E_i}}{\sum_{j}^{} ...
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2answers
179 views

Uniform distribution on the n-sphere.

I have the next RV: $$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$ where $$X_i \tilde \ N(0,1)$$ It's a random vector, and I want to show that it has a uniform ...
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1answer
2k views

Joint distribution of multiple binomial distributions

In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them. The original file can be ...
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3answers
283 views

Compute value of $\pi$ up to 8 digits

I am quite lost on how approximate the value of $\pi$ up to 8 digits with a confidence of 99% using Monte Carlo. I think this requires a large number of trials but how can I know how many trials? I ...
0
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1answer
82 views

Gaussian expectation of an exponential function

I am struggling to prove this, $$ \int \mathcal{N}_\mathbf{x}(\mu,\Sigma)e^{a^T\mathbf{x}}d\mathbf{x} = e^{{a^T\mu}+\frac 12a^T\Sigma a} $$
0
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1answer
106 views

expectation of logarithm under generalised inverse gaussian

I want to follow the following integral: $$\frac{1}{C}\int_0^\infty \log(z)\,z^{p-1}\exp\left(-\frac{az+b/z}{2}\right)\,dz$$ where C is the normalising constant. The following might be useful ...
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2answers
4k views

Fisher information of a Binomial distribution

The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
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1answer
830 views

Approximating a Poisson distribution to a Normal distribution

I have the following problem I'm trying to solve: I know that the quantity of complains in a call center is a Poisson variable with $\lambda=18 $ costumers/hour, and that the probability of being ...
2
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1answer
304 views

Fast generation of Pareto-distributed randoms.

I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
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0answers
96 views

Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin

I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
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2answers
57 views

Sample $x$ from $g(x)$

I got confused with all this randomness and probability functions. I was trying to implement the rejection sampling method which (apparently) is really simple. I was reading from Rejection Sampling in ...
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4answers
279 views

Independent and uniformly distributed on $(\frac{1}{2},1]$

I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to ...
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0answers
49 views

please show that $\hat\mu_i\sim N(\mu_i,\frac {\sigma^2}{n_i})$

Statistical model for Complete Randomized design $y_{ij} = \mu + \tau_i + \epsilon_{ij}$ where, $i$ denotes treatment and $j$ denotes observation. $i=1,2,...,k\quad and \quad j=1,2,..., n_i$ ...
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1answer
670 views

Accept reject method to generate random numbers

The method says that having a proposal $g(x)$ Sample $X^* \tilde ~ g(x)$ and $U \tilde ~ Unif(0,1)$ Accept $X = X^*$ if $U ≤ f(X^*) / M g(X^*)$ Moreover, $M$ is constant that satisfies $Mg(x) ≥ ...
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1answer
584 views

Why is a CDF right-continuous at “a” in [a,b), when property Pr(a<X≤b) doesn't even require point “a” to exist, and “b” could carry baggage?

c.f. wikipedia:Cumulative distribution function properties "Every cumulative distribution function F is (not necessarily strictly) monotone non-decreasing (see monotone increasing) and ...
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2answers
40 views

Central limit theorem - std dev away from mean

I was reading about the CLT and found something that I think people use interchangeably. On one hand I found that 68% of the means are 1 standard deviations from away and 95% are 2 std dev. On the ...