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

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9 views

Bayesian update and discontinuous cdf

I have an uniform random variable $x\sim U(0,1)$. I receive a signal $z$ about $x$ that is given by $$ z=y(x)+\varepsilon $$ where $\varepsilon\sim U(-\frac{1}{2},\frac{1}{2})$ (independent from $x$) ...
2
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3answers
56 views

How do I calculate dice with addition and subtraction based on dice rolls?

I am trying to figure out how to calculate results on a group of dice where some results are positive and others are negative. Example: I roll a group of dice that are fair and six-sided. Each roll ...
0
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0answers
12 views

A simple PDE related to symmetric functions

This question was posed on mathoverflow but was put on hold. So far nobody has been able to give any hint either to the solution or to why it is trivial. Disclaimer: as far as I can tell this is not ...
0
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0answers
18 views

Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
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1answer
21 views

The probability to log on a computer from a remote terminal is 0.7.

The probability to log on a computer from a remote terminal is $0.7$. Let $X$ denote the number of attempts that must be made to gain access to the computer. Find: (a) The distribution of $X$ and ...
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1answer
16 views

Distribution of a random variable to the power of 3

I have to answer the following question: $X$ has a uniform distribution between 0 and 1. What is the distribution of $X^3$? I'm not looking for an answer, just want to know how I should begin to ...
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3answers
53 views

Does $n/\Sigma_{i=0}^n(1/X_i)$ converge to $0$ in probability for $X_i$ iid standard uniformly random variables?

Suppose $X_i \sim\operatorname{uniform}[0,1]$ and that they are iid. Does $n/\Sigma_{i=0}^n(1/X_i)$ converge to $0$ in probability? A simulation seems to indicate that it does. But as the expected ...
2
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1answer
19 views

Exponential law with both positive and negative values

The exponential law with density $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ and $f(x)=0$ for $x < 0$, is well-known. What's the name of the distribution which has $$f(x) = \frac{1}{2} ...
1
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0answers
10 views

Find expected step, on which half-normal RV exceeds a scalar value?

I have defined a following problem. Given is a non-negative integer variable (steps) $s\in[0,1,...)$, and a scalar random variable as a function of $s$, $R(s)$. Random variable is half-normally ...
8
votes
3answers
310 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
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0answers
16 views

Necessary and sufficient conditions (1) rv to density function (2) distribution to rv

(1) Let $(\Omega,\mathcal{F},P)$ be a probability measure space and $X:\Omega \rightarrow \mathbb{R}$ a random variable. Let $P_X,~F_X$ denote the probability measure, pdf induced by $X$, ...
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1answer
35 views

Multivariate normal value standardization

I am wonder how to standardize multivariate normal value. Normal standard multivariate distribution of $q$ variables is $z\sim N_q(0_q,I_q)$. I have found that $Bx\sim N_q(Ba,B\Sigma B^T)$ and based ...
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0answers
9 views

Comparing distributions against expected to determine the one fitting better

I have a 4 sets of observed absolute frequencies for a categorical variable and the expected frequence for each category (not normal distributions). Would it be correct to use the Chi-square goodness ...
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1answer
24 views

The product of multiple univariate Gaussians

What is the final result of $$I=\mathcal{N}_{x}(\mu_1,v_1)\,\mathcal{N}_{x}(\mu_2,v_2)\ldots\,\mathcal{N}_{x}(\mu_n,v_n)=\frac{1}{\sqrt{2\pi\,v_1} } e^{ -\frac{(x-\mu_1)^2}{2v_1} } ...
0
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0answers
5 views

Difference between the SCM converging to the Marcenko-Pastur distribution and Johnstone's result about the top eigenvalue

I have a confusion which I suppose must be rather basic. As I understand, in the 60s/70s it was known that the empirical eigenvalue distribution of the sample covariance (of $n$ i.i.d. standard ...
1
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1answer
183 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
0
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1answer
135 views

Distribution in Polya's Urn / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
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0answers
10 views

Constructing a copula that satisfies the desired condition

Exercise 2.8 in Roger Nelson's An Introduction to Copulas asks the reader to construct a copula $C(u,v)$ not equal to $\max(u + v -1 , 0)$ that satisfies the property $$ C(u,u) = \max(2u - 1,0) $$ ...
0
votes
1answer
28 views

Predictive Distribution with Normal Prior

Given $\Theta = \theta$, let $X_1, X_2, \dots, X_n, X_{n+1} \sim \mathcal{N}(\theta, \sigma^2)$ be independent. $\Theta \sim \mathcal{N}(\theta_0, \tau^2)$. What is the easiest way to find the ...
0
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1answer
21 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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1answer
946 views

PDF of the ratio of two independent Gamma random variables

Let $X \sim \operatorname{Gamma}(a,\lambda)$ and $Y \sim \operatorname{Gamma}(b,\lambda)$ being independent. Find the PDF of the ratio $W=X/Y$. I found $$ f_W(w) = \frac{\Gamma(a+b)}{\Gamma(a) + ...
5
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0answers
62 views
+250

Finding where the tail starts for a probability distribution, from its generating function

Suppose we generate "random strings" over an $m$-letter alphabet, and look for the first occurrence of $k$ consecutive identical digits. I was with some effort able to find that the random variable ...
3
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1answer
471 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 ...
1
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3answers
55 views

An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon.

An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon. For a bowl, one chooses at random five kinds (not necessarily different). $(a)$ How many different bowls can be made? ...
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1answer
40 views

A box with $3$ types of colored balls.

In a box there are $15$ white balls, $8$ black balls, and $12$ red balls. We extract $6$ balls, without putting them back. $(a)$ What is the probability that the first ball is red, the second and ...
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1answer
31 views

Calculate the density of $X=X_1*X_2$ using dirac function.

Let $X_1$ have p.d.f $$p_1(x_1)=\gamma^2x_1 \cdot \text{exp} \left( \frac{-x_1^2}{2} \right),$$ and $X_2$ have p.d.f $$p_2(x_2) = \frac{1}{2 \pi} \text{exp} \left( \frac{-x_2^2}{2} \right). $$ ...
0
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0answers
26 views

How to find out the following probability?

I need to find $\mathbb{E}_d[\mathbb{P}\left\{X\le\mu\right\}|\hspace{1mm}d]$ with \begin{equation} ...
0
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0answers
13 views

Distribution of maximum frequency in uniform sample

If I take $n$ random integers from $1$ to $m$, how do I calculate the distribution of the number of occurrences of the most frequent number? Any hints or initial approaches? I thought to get the ...
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2answers
24 views

Distribtution of the maximum of three uniform random variables.

How do I get the cumulative density function of $Y$? $X$ is a continuous random variable with pdf $$f(x) = 1,\quad 0 < x < 1. $$ Three independent observations of $X$ are made. Find the pdf ...
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0answers
19 views

Probability of maximum degree in On random graphs I by Erdos

In The Maximum Degree of a Random Graph by RIORDAN et al., the authors commented that the study of the distribution of the maximum degree $d^{max}(G)$ of a random graph $G$ was started by Erdos and ...
0
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1answer
31 views

Confused with the power set of an integer

I am going through The Maximum Degree of a Random Graph by RIORDAN et al. On the second page, the notation $\mathbb{P}(\mathcal{D})$ is used which I assume the power set of the set $\mathcal{D}$. ...
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0answers
13 views

Estimation of binomial probabilities $f(r)$ over $r \in [0,\frac{1}{2}]$

I want to fit a (decreasing) univariate function, \begin{equation} f(r), \end{equation} over $r \in [0,\frac{1}{2}]$ to a series ($r =\frac{1}{100}, \frac{2}{100}, \frac{3}{100} ,\ldots,\frac{1}{2}$) ...
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1answer
20 views

Gaussian RV distribution [on hold]

Prove, for $X$ and $Y$ independent zero-mean Gaussian random variables with variance $σ^2,$ that the distribution of $Z=\sqrt{ X^2+Y^2}$ is Rayleigh distributed and that the distribution of $Z^2$ is ...
2
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0answers
16 views

Prove using moment generating function: $\mathbf{y}^T \mathbf{A} \mathbf{y} \sim \chi^2(a) \iff \mathbf{A}^2=\mathbf{A}$ and rank($\mathbf{A})=a$.

I'm trying to prove the following using the moment generating function: For $\mathbf{y}\in \mathbb{R}^n \sim \mathcal{N}(0,I_n)$, one has $\mathbf{y}^T \mathbf{A} \mathbf{y} \sim \chi^2(a) \iff ...
0
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0answers
18 views

Entropy of sum of uniform random variables on a simplex [duplicate]

For two i.i.d random variables $X$ and $Y$, which are uniformly distributed on the $n$-dimensional simplex $\Delta_n= \left\{(x_1,\ldots,x_n): x_i \geq 0, \sum_i x_i \leq 1 \right\}$, I want to find ...
2
votes
2answers
29 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...
0
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1answer
32 views

Probability Distribution: Verification of my Thinking

More than anything, I just need someone to confirm for me that I'm on the right track. So I have a table that has some random variable $X$ which has a probability distribution table of: ...
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0answers
13 views

What star rating is representative of this distribution? [on hold]

100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 = 33, 2 = 26, 3 = 12, and 4 = 28. What star rating would you say is "representative" of these 100 people: 2.36 (2), the average, ...
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votes
1answer
23 views

Waiting time for two independent poisson processes

Order of Events in Poisson Processes Assume that you have two independent Poisson process, $N_1(t)$ with rate $\lambda_1$ and $N_2(t)$ with rate $\lambda_2$. The probability that $n$ events occur ...
5
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0answers
35 views

markov chain: 2 state chain

I have a machine. It has two states, broken or working. If it is working, then it will be broken with probability $q=0.1$. If the machine is working, I will make \$1000 dollar a day. If it is broken, ...
0
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1answer
38 views

Poisson probability of an event A before event B

I'm trying to calculate the probability of two poisson processes events happening one before the other, with two different $\lambda$s. The way I see it, I can word it as the probability of event $A$ ...
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0answers
8 views

Derive distribution of a random variable given an observed perturbation

I have a process by which some initial value $x_0$ is perturbed by $\epsilon$ to $x_{obs}$, where $\epsilon$ is a random number drawn from a PDF $p(\epsilon)$. Given a particular observed value ...
1
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3answers
35 views

probability density function chi squared

Exercise I've been tasked with deriving the probability density function for a chi-squared random variable $$f(x;q) = \begin{cases} \hfill 0 \hfill & x\leq 0 \\ \hfill ...
1
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1answer
36 views

Expected value of a poisson process

I've been searching for a while but I can't seem to figure out how to find the expected value of a poisson process up to an arbitrary time. Let {$N(t),t≥0$} be a Poisson process with rate $λ$. How ...
1
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0answers
21 views

maximum likelihood estimators of a shifted gamma distribution?

i had this question in my exam but didn't know how to solve this apart from constructing the likelihood function and differentiating .but got stuck in the middle of nowhere.please help . the answer ...
2
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0answers
27 views

Distribution of distinct object problem

So i was given this question. How many ways are there to place 10 distinct people within 3 distinct rooms with exactly 5 people in the first room and 2 people in the second room? So i asked this ...
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0answers
14 views

geometric and exponential distributions

A link with transmission rate $R_b[bit/sec]$ is used to forward packets having random size $l[bit]$ which has a geometric PMD: $p_l(k) = p(1-p)^{k-1}$ Prove that, if $E[l] = \frac{1}{p}$ is large ...
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0answers
6 views

Centered Poisson, Scaled Poisson, Transformed Poisson

Given $y_1,y_2,\ldots,y_N$ with $y\sim \operatorname{Poisson}(\lambda)$. The question is, what is the distribution of $y_i-\bar{y}$ and $\frac{y_i-\bar{y}}{\bar{y}}$, where $\bar{y}=\sum_1^N y_i/N$. ...
1
vote
1answer
21 views

Expectation of scaled sum of squares of iid random variables

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. What is ...
0
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0answers
15 views

Moments of censored exponential distribution

I have a question as to whether my calculation of moments of censored exponential distribution is correct. I have two random variables $T_A=\min(\tau,t_1)$ and $T_B=\min(\tau,t_2)$, where $t_1<t_2$ ...