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

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9 views

Entropy of sum of uniform random variables on a simplex

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
28 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
12 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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0answers
25 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
20 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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6 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
33 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
34 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 ...
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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 ...
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0answers
26 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
10 views

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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12 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
14 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
19 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 ...
1
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1answer
19 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
9 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
12 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
36 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.
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1answer
48 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
52 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 ...
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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 ...
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0answers
42 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
16 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 ...
4
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1answer
22 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
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0answers
12 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
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2answers
22 views

Find PDF on $[0,6]$ such that $P([1,3]) = 0.5$

Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$ That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} ...
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0answers
12 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
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0answers
23 views

What is expected value of only positive numbers [duplicate]

Normal distribution with mean zero and standard deviation 1. What is the expected value of just the positive variables
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2answers
44 views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
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1answer
17 views

Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
3
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3answers
59 views

Upper bound for difference of Poisson random variables

Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then (i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq ...
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2answers
52 views

Methods for calculating the mean and variance of a distribution created from the addition of two normally distributed quantities

I'm trying to understand how to interpret the following which refers to determination of the mean and variance of a distribution that's the result of adding two normally distributed random variables. ...
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0answers
13 views

Position error probability distribution when distance and angle error distributions are zero mean Gaussian

In one problem we are estimating the position of an object from the measurement of its distance $\mathbf{r}$ from a point as well as its angle $\mathbf{\theta}$ from the reference direction. The error ...
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0answers
27 views

Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts $(2,3,5)$. So if you buy a ticket you can win a pot which will payout your ticket price multiplied by one of those numbers. The organizer expects a margin profit ...
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2answers
17 views

Estimate on Probability of a standard normal variable

In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result: If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P ...
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1answer
16 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
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0answers
8 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
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2answers
77 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
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1answer
10 views

Expectation of the minimum of two $\mathcal U(0, 1)$ r.v.'s conditional on it being greater than or equal to some value

Let $X_1, X_2$ be i.i.d. $\mathcal U(0, 1)$ (continuous) r.v.'s, and let $0 \le R \le 1$ be some number. What is $\mathbb E[\min(X_1, X_2) \mid \min(X_1, X_2) \ge R]$? My attempt: Let $Y = ...
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2answers
29 views

Convergence of Sum of Random variable to another - Cantor function

Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this ...
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0answers
10 views

Sampling Distributions. Statistics [on hold]

I'm stuck in this problem: Problem Picture I did the literal a and b, but the rest of them I don´t understand, the reason why.
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0answers
33 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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2answers
34 views

What probability mass, density or distribution function might corresponds to this moment generating function? [duplicate]

I have somehow come up with a random variable $X$ with moment generating function (assuming it exists) $$M_{X}(t) = -t (1 - e^t)$$ What is the probability mass, density or distribution function? It ...
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47 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
2
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1answer
21 views

Limiting Distribution $\Delta-$method

Let $Y_n\sim \chi^2(n)$. What is the limiting distribution of $U_n= \dfrac{\sqrt{Y_n}-\sqrt{n}}{\sqrt{2}}?$. What I know is that if $X_i\sim \chi^2(1)$, I can write $Y_n = \sum\limits_{i=1}^n X_i$. ...