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

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3
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1answer
56 views

Proving a Variation of the the Central Limit Theorem

I am trying to prove the following: Let $X1, X2, . . .$ be positive, i.i.d. r.v.s with mean $\mu$ and finite variance $\sigma^2$, and let $S_n = \sum_{k=1}^{n} X_k$ , $n \ge 1$. Show that $\frac{S_n ...
2
votes
0answers
23 views

Distribution of $f(x,|h|)$, being $|h|$ rayleigh distributed

INTRODUCTION Let's supose we receive the following signal: \begin{equation} y[n] = hx[n]+W[n] \end{equation} where: $x[n] = Ae^{j2 \pi f_c t}$ is the transmitted signal $f_c$ is the carrier ...
1
vote
2answers
43 views

Confidence interval for Poisson distribution coefficient

This is an exam question, testing if water is bad - that is if a sample has more than 2000 E.coli in 100ml. We have taken $n$ samples denoted $X_i$, and model the samples as a Poisson distribution ...
2
votes
1answer
30 views

The asymptotic equivalence of LR, Wald and score tests

Suppose that $Y_1, \ldots, Y_{n}$ are iid from a Bernoulli distribution with parameter $p$ and consider $H_0 : p = p_0\,.$ The test statistics are $$ T_W = \frac{n ({\widehat p} - p_0)^2}{{\widehat ...
0
votes
1answer
35 views

Simulating Random Vectors Based on Conditioning

I'm working on a project where I need to simulate random vectors $(Y, X_1,\dots,X_n)$ in order to understand the joint distribution $f(y,x_1,\dots,x_n)$. I wish to simulate enough random vectors so ...
2
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4answers
84 views

What does this definition mean: $F_Y(y) =P(Y<y)$?

I am doing calculations on $F_Y(y) := P(Y<y)$, but I am clueless as to what $P(Y<y)$ means. For instance the following question: Given function: $f_X(x)= 2\lambda x e^{-\lambda x^2}$ when $x ...
0
votes
3answers
19 views

Combined Binomial Distribution Problem.

I have the following problem: 70% of women respond positively to a test, while only 40% of men do so. If 10 participants are selected (5 women and 5 men), what is the probability that only 1 man ...
2
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0answers
37 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
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0answers
46 views

What is the product of two independent random variables (as mentioned below)?

Let $X$ and $Y$ be two random variables with: $\begin{equation} f_{X}(x) = \begin{cases} e^{-\lambda T} & \text{if } x = 0;\\ \lambda T e^{-\lambda T(1-x)} & \text{if } 0 < x \leq ...
1
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0answers
149 views

Linear programming: constraints that depend on sign

Edit: following a comment, more detail and context, and removed lengthy confusing remains of previous edits I basically want to check whether there is a sequence $y_1,\dots, y_n \in (-\infty,0]$ that ...
0
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1answer
16 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
1
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0answers
23 views

How to solve for a Phase Function Cumulative Distribution function (CDF) calculation give a pdf …

I am attempting to solve for the CDF (more specifically the inverse CDF, but that is easy once I have the CDF) - Cumulative Distribution function given a Probability Distribution Function (pdf) and g ...
4
votes
1answer
32 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
3
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0answers
21 views

Expected value of sorted subsequence

Consider the following discrete random variable: Given an array of size n containing random unique integers, what is the maximal length of a sorted subsequence from that array. What is the expected ...
5
votes
1answer
81 views

Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j ...
3
votes
1answer
37 views

Jacobian Transformation p.d.f

Suppose $X$ and $Y$ are continuous random variables with joint p.d.f. $$f(x,y) = e^{-y},\,\, 0<x<y <\infty$$ (a) Find the joint p.d.f. of $U=X+Y$ and $V=X$. Be sure to specify ...
0
votes
1answer
18 views

Linear combination of gaussian variables

If $X\sim N(0,\sigma_1^2)$,$Y\sim N(0,\sigma_2^2)$ and given that X,Y are independent random variable with normal distributions, then for the random variable $U=\alpha X+\beta Y\sim N(\mu,\sigma^2)$ ...
1
vote
1answer
44 views

Is it possible to determine if a process is random

Imagine the following experiment: someone is sitting behind the screen and calls out a sequence of numbers: "1! 3! 5! 3! 4! ...". Let's say he/she and I agreed beforehand that all numbers are ...
3
votes
3answers
59 views

Can a finite data set have all its values within $n$ standard deviations from the mean?

Aside from the trivial $(x,x,x,x,...)$ data set, is it possible to have all the elements of a data set within some $n$ standard deviations from the mean? What is the minimum possible value for $n$ ...
2
votes
1answer
46 views

Unable to understand what kind of pdf and its origin

I am facing difficulty in identifying how the formula given by Eq(2) in the paper Wen-Chi Tsai and Anirban DasGupta, On the Strong Consistency, Weak Limits and Practical Performance of the ML ...
0
votes
1answer
28 views

Uniform distribution on sphere

Let $U = (u_1, u_2, u_3)$ is random vector uniformly distributed on unit sphere $S^{2} \subset \mathbb{R}^3$. Are $u_1, u_2, u_3$ mutually independent ? I guess not, but I have no idea to prove it.
0
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0answers
32 views

Expected probability maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like $$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$ Here $f(k,x)$ is actually a probability coming from a ...
-1
votes
1answer
108 views

Conditional Expected Value of Product of Normal and Log Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} E\left[\left.\left(e^{X}Y+k\right)\right|\left.\left(e^{X}Y+k\right)>0\right]\right. \end{eqnarray*} ...
0
votes
0answers
35 views

What is the probability of a collision (birthday problem) using a Zipf distribution?

Assuming a random number generator gives one of n numbers using a Zipf distribution rather than a uniform distribution, what is the probability of each number being a number that was already generated ...
0
votes
1answer
22 views

Properties of a distribution function

I'm having trouble understanding the properties of a distribution function. My book only gives these short rules. http://www.pixhost.org/show/2720/28297379_2015-06-22-15-27-44.jpg My professor said ...
1
vote
1answer
17 views

Transformation of two i.i.d. uniform random variables

G'day folks, I'm trying to work through a problem in preparation for an exam and it's got me stumped. The question is: Let $X_{1}$ and $X_{2}$ be i.i.d. $U(0,1)$ random variables. Let ...
2
votes
1answer
78 views

Median of Poisson Distribution

I've just taken Probability and Statistics exam and here is the question that becomes a hot debate among my friends in the classroom: What is the median of a random variable that follows a Poisson ...
1
vote
1answer
28 views

Default positive/(non-negative) probability distribution

So if you have a random variable that corresponds to a natural phenomenon and you don't know how it is distributed, you often assume it is normally distributed. Now I have a random value that I know ...
0
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2answers
41 views

Explaining the form of the Gaussian measure

The Gaussian density $\mu(dx)=e^{-x^2/2}\ dx$ is fundamental in probability theory. Does anyone have a (non-computational) heuristic why this function should be special? (By non-computational, I mean ...
0
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0answers
13 views

Discrete grid: random points with radial probability distribution

I have a cubic 3D grid of $N^3$ points. I randomly choose a certain point to be the centre. Now I want to generate random points which obey a certain probability distribution $\rho(r)$ which depends ...
1
vote
1answer
36 views

Application Problem: Random Sums of Random Variables and Correlation

I am trying to answer the following: The number of traffic accidents per year at a given intersection follows a Poisson(10, 000)-distribution. The number of deaths per accident follows a ...
0
votes
1answer
28 views

Determining Probability Distribution from Probability Generating Function

I am trying to solve the following: The nonnegative, integer-valued, random variable X has generating function $g_{X(t)}=log(\frac{1}{1-qt})$. Determine $P(X = k)$ for $k = 0, 1, 2, . . .$, $E[X]$, ...
0
votes
3answers
50 views

An urn contains 12 red marbles and 10 blue marbles.

We draw five marbles from this urn without replacement. Given that the first two marbles are blue, what is the probability that there is at least one blue marble among the next three marbles? I'm ...
1
vote
2answers
39 views

Roll a pair of dice

(a) What is the probability of rolling at least 9? I draw the table and I came up with and answer of 5/18 from adding P(9)+P(10)+P(11)+P(12) (b) if one die rolls a 4, What is the probability of ...
1
vote
0answers
40 views

Alpha, Bravo, Charlie, and Delta are playing in a tennis tournament.

(a) If there are no ties at the end of the tournament, how many different rankings are possible for the four players? (b) If all the players have the same skill, what is the probability that Charlie ...
0
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0answers
24 views

Bhattacharyya Distance of Age/Gender Groups?

I'm calculating distances for groups based on Age/Gender Compositions (to rank their similarity in demographic composition.) I'm working with the following: Men 18-34, Men 35-49, Men 50-64, Men ...
0
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0answers
33 views

“Distance” between two distributions

Hello, I need help understanding this problem. I have no idea how I'd approach it and have some problems following the solution. I can understand the first line, how $F_1$(x)=x because it increases ...
3
votes
1answer
84 views

Uniform distribution on a sphere

Consider the unit ball $S_n$ (centered at the origin) in $\mathbb{R}^n$ for $n \ge 1$ and a stochastic process $(X_t)_{t\ge 0}$ taking values in $\mathbb{R}^n$. Let $T = \inf\{t > 0 \colon X_t ...
0
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0answers
10 views

Convergence to Gumbel distribution of $\sum_{i=1}^n\left(\frac{Y_i-1}{i}\right)$ where $Y_i\sim$Exp$(1)$

Let $(Y_n)_n$ be a sequence of independant r.v.s exponentially distributed with parameter 1. Let $M_N:=\sum_{i=1}^n\frac{Y_i-1}{i}.$ There are two parts of the question - first find the variance of ...
0
votes
3answers
65 views

Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity ...
12
votes
1answer
174 views

About the “Cantor volume” of the $n$-dimensional unit ball

A simple derivation for the Lebesgue measure of the euclidean unit ball in $\mathbb{R}^n$ follows from computing $$ \int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx $$ in two different ways. See, for instance, ...
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2answers
52 views

We draw 5 cards from a standard deck of 52 cards without replacement.

We draw 5 cards from a standard deck of 52 cards without replacement. Find the probability of drawing all the cards of the same suit. I think the answer is P of same suit = ...
0
votes
1answer
40 views

conditional probability combining discrete and continous random variables

Let us define $Y$ a continuous random variable having density $f(y,\theta)$ and X a discrete random variable such that $X=\mathbf{1}_{ \{Y \in [a_{i-1}, a_{i}) \}}$ . I want to compute the ...
0
votes
1answer
23 views

Given the probability density of random variable $X$, what is the density of $Y=aX+b$?

I have a random variable $X$ with probability density $f_X$ and want to determine the probability density $f_Y$ of $Y=aX+b$ with $a,b \in \Bbb R$. How do I proceed here?
1
vote
1answer
50 views

Query concerning $\int\limits_0^\infty {{e^{\left( {it - 1} \right)x}}dx} = \frac{1} {{1 - it}}$

$$ \int\limits_0^\infty {{e^{\left( {it - 1} \right)x}}dx} = \frac{1} {{1 - it}}.$$ The above came up in a probability question and I was fairly happy it is true, I just don't really feel I ...
0
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1answer
9 views

With $e_i\sim\exp(1)$ why does $\prod_{j=1}^n\exp(ite_j) = (\frac{1}{1-it})^n$?

We have $S_n=\sum_{i=1}^ne_i$ with $e_i\sim\exp(1)$ why does $\prod_{j=1}^n\exp(ite_j) = \left(\frac{1}{1-it}\right)^n$? I just want to understand the following line from my notes and hope it is ...
0
votes
1answer
22 views

For given mean $\mu$ of random variable X in [0,1], what is the probability distribution function $p(X)$ that makes $VAR(X)$ maximum?

Given the conditions $\int_{0}^{1} p(x)dx=1$, $\int_{0}^{1} xp(x)dx=\mu$ and $p(x)\ge0$ for $\forall x \in [0,1]$, What probability distribution function $p(x)$ makes $Var(X)$=$\int_{0}^{1} ...
0
votes
1answer
75 views

Calculate the maximum probability of a result of rolling n dice of varying number of faces

Disclaimer: I'm a computer programmer more than a mathematician, so reading text like that of the answer to this question is a little (read: a lot) over my head. I've written an algorithm ...
0
votes
1answer
25 views

From pairwise P(A > B), to P(A > all distributions in set)

$\{D_0,…,D_n\}$ is a finite collection of independent (but not necessarily identically distributed) random variables. Define $f(x,y)=P(D_x≥D_y)$ and $g(x)=P(∀y:D_x≥D_y)$. Does $f$ determine $g$, and ...
0
votes
1answer
25 views

Probability of a message being processed in a given interval

Let's say I have a queue and a server that connects to the queue in an interval basis and process all the available messages. The server connects, tries to get a message, if there is one then server ...