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

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9 views

Suppose $X_i$ are iid $\operatorname{uniform}[0,1]$ distribution, does $1/\sum_{i=0}^n(1/X_i)$ converges to $0$ in probability?

Suppose $X_i$ are iid $\operatorname{uniform}[0,1]$ distribution. Does $$\frac{1}{\sum_{i=0}^n(1/X_i)}$$ converges to $0$ in probability? The simulation seems it does. But as the mean of $1/X_i$ ...
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1answer
15 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} ...
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2answers
40 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 ...
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0answers
13 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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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 ...
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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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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 ...
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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 $$ C(u,u) = \max(2u - 1,0) $$ ...
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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 ...
0
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1answer
25 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 ...
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1answer
33 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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3answers
45 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
28 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 $$p_2(x_2) = \frac{1}{2 \pi} \text{exp} \left( \frac{-x_2^2}{2} \right). $$ ...
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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
12 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
23 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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1answer
20 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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0answers
18 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
12 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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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 ...
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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 ...
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0answers
17 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 ...
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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, ...
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, ...
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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 ...
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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 ...
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1answer
37 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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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 ...
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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42 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 ...
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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 $\bar{y}=\sum_1^N y_i/N$. ...
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0answers
13 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
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 $t_1<t_2$ ...
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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 ...
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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 ...
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0answers
11 views

Invariant distributions: Applications in the real World

I'm studying about invariant distributions for Markov processes; say in the context of dynamics of Random Neural Networks (biological Networks). I can't fully understand what does an invariant ...
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0answers
9 views

Derivation of variance of a linearly transformed vector

I am trying to derive the variance of a linearly transformed vector. A result was given here. $$ \mathbf{y} = X \, \mathbf{b} $$ $$ \mathbf{b} \sim \mathcal{N}( \mathbf{0}, \sigma^2 I) $$ If we say ...
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1answer
33 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
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0answers
19 views

Probability and Statistics [on hold]

How can I check if a Moment Generating Function is valid or not? I tried using the definition for that but it didn't help me at all.
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0answers
37 views

Joint probability distribution $X, Y$.

$f(x,y)= \frac{3}{2}(x²+y²)$, $\:\:0 \leq x,\: y \leq 1$ $0$, elsewhere Determine whether or not $X$ and $Y$ are independent. Independent characteristic when $f(x,y)=f(x)f(y)$ To find f(x) and ...
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1answer
23 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
2
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1answer
49 views

Find a continuous PDF on $[0,6]$ for given probabilities

Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$. After some calculations I came up with ...
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3answers
53 views

Find the distribution of $Y = -\log (1-X)$ given that $X\sim U(0,1)$.

If $X \sim U (0,1)$ then if we define a new random variable $Y=-\log (1-X)$ then what will be distribution of $Y$. Please explain.
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0answers
26 views

Appropriate distribution for set of probabilities $p_1 ,…, p_n$

I am doing some evaluation of a system, that has set of probabilities $p_i$ $i= \in \{1,...,N\}$, I need to model them as random variables such that : $$ \sum_i p_i \leq 1$$ and $$ 0 \leq p_i \leq 1 ...
2
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0answers
32 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
1
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0answers
44 views

Distribution and expectation value of ceiling function of Poisson

There is Poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is less than 1). How can I ...
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0answers
17 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...