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

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Two-group Probability question

I had this question on a test a few days ago, and when I got the test back, this question was marked wrong: In a group of 500 people, 60% of them are female. In this same group, 10% of the people ...
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0answers
53 views

Generalised Poisson Distribution

While studying stochastic processes (specifically the paper http://arxiv.org/abs/cond-mat/0412129v1) I have come across a probability distribution that is a generalisation of the standard Poisson ...
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1answer
75 views

Find conditional probability $\mathbb{P}(X \le x | \max(X,Y)) $

Let $X,Y$ be iid such that $X\sim F>0$ and $Y \sim F>0$ ($X$ and $Y$ have the same probability distribution). Find $\mathbb{P}(X \le x | \max(X,Y)) $. I know that $\max(X,Y) \sim F^2$. I ...
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1answer
29 views

Does the specific bias random coins determines whether functions of these are independent or not?

Consider 3 independent r.v.s $X_1$, $X_2$, $X_3$ that represent the outcomes of three (independent) fair coin tosses. Let 1 denote heads and 0 denote tails. Let two new random variable be defined ...
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1answer
17 views

Does a constant of integration changes the shape of a distribution?

Let $f(x)$ be the frequency distribution of the variable $x$. Let assume that $\int^{\infty}_{-\infty} f(x) ≠ 1$. Let $g(x) = C f(x)$ such as $C$ is the constant of integration so that ...
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1answer
88 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
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1answer
20 views

Explanation of this situation with two random variables - $X$ conditionally distributed on $N$?

Let $N$ have a Poisson distribution with parameter $\lambda = 1$. Conditional on $N = n$ let $X$ have a uniform distribution over the integers $0, 1, ..., n+1$. What is the marginal distribution of ...
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1answer
143 views

how many types of events actually exists in the theory of probability?

I read many article on the internet and found that there are only three types of event that can be occurred(or that has been considered in the probability theory). those are : mutually exclusive ...
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1answer
107 views

ratio distribution of gamma with different parameter

Let $X$ be gamma distributed random variable with parameters $a$ and $b$. Let $W$ be gamma distributed random variable with parameters $c$ and $d$, such that \begin{equation} f_X(x) = ...
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0answers
243 views

Extinction probability of binomial branching process tends to poisson one.

The folowing is stated and proved in the random graphs book by Luczak, Janson, Rucinski and this is on page 108 in the Giant component section. I can't understand why the conclusion follows from the ...
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113 views

Square Root Transformation on a Random Variable

I read in my textbook that if Y ~ N(0,1) and Z is exponential with mean 1 (f(x) = exp(-x)), then X ~ sqrt(2Z)*Y follows a double exponential (Laplace) distribution with parameter 1 (f(x) = 1/2 ...
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21 views

random variables about birthdays

So lets do away with months/days and assume everyone has a birthday $X_k$ which corresponds to a number from $1$ to $365$, uniformly distributed In a group of $n$ people, let $M=\max (X_k)$ be the ...
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1answer
56 views

Expected time of n events to complete

What is the expected time do i have to wait until n events are completed each distributed in exponential time $\mu$? I thought that $1/\mu$ is the expectation for the first since the events are ...
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1answer
31 views

Jump Set v. Range of Randome Variable

What is the difference between the range of a random variable X, and its jump set? I know that they are not equivalent sets, e.g. for a continuous RV, the range is $(- \infty , \infty)$, but the jump ...
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2answers
69 views

Distribution of a distance between random numbers

I'm working on a problem in which I came to a question concerning distribution law of a result of operations on random variables. I will ask a simple question and hope to understand the concept from ...
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0answers
96 views

Mean of the Multivariate Wallenius Non-Central Hypergeometric Distribution

An urn contains $N$ balls where ball $i$ is of size $w_i$. We draw $n$ times without replacement. Let $x_i$ be the random variable indicating whether the ball $i$ has been drawn ($x_i=1$) or not ...
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2answers
111 views

What is the sum-capacity for a non-symmetric interference channel for information theorists?

This question is dedicated for people who are experts in information theory. An interesting result for a two user interference channel in information theory, is the sum-capacity to within one bit. It ...
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1answer
156 views

A Law of Large Numbers Without Replacement

Let $(n_1,...,n_r)$ be $r$ positive integers, and let $n=n_1+...+n_r$. Fo each positive integer $m$ consider an urn containing $mn$ balls, of which $mn_1$ are of type 1,..., $mn_r$ of type r. For each ...
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1answer
168 views

Expected Values probalem

The problem is: In a world without gravity, a very small gun shooting point-like balls is located at the lower left end $(0, 0)$ of a $2D$ corridor. The corridor has length $L = 100\thinspace m$ and ...
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0answers
51 views

Conditional mean: E(Y|x)

Please help.I am not sure with my answer.Anyways, the problem goes this way: Find the conditional mean of $Y$ given $X=x$ ,$E(Y|x)$, if X and Y have the joint pdf of $f(x,y)=21x^2y^3, ...
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1answer
48 views

Probability of $P(X_1<X_2|X_1<2X_2)$

Find the $P(X_1<X_2|X_1<2X_2)$ Given: $$f(x) = e^{-x}, \qquad 0<x<\infty$$ zero elsewhere. The rvs have same pdf and they are independent variables. Here is my attempt: ...
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1answer
27 views

Distribution function and probability of random variable R with density function $f(x) = 1/2e^{-|x|}$

So we have a random continuously variable $R$ with density function $f_R(x) = 1/2e^{-|x|}$. First I need to sketch the distribution function of $R$. So $$F_R(x) = \int_{-\infty}^x f_R(x) dx$$, but ...
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2answers
63 views

How is the binomial distribution connected with the theoretical approach to probability?

I've been told the theoretical approach to probability is defined as follows $$\operatorname{Pr}(\textsf{something})=\frac{\textsf{Favorable events}}{\textsf{possible events}}$$ This has to be ...
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2answers
132 views

Conditional expectations of $E(X+Y|z)$

Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere.I was instructed to determine the cumulative df of $x,y,z$. Here is my answer $$F (x,y,z) = ...
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1answer
44 views

Probability of $P(X_1 X_2\le 2)$

What is the probability of $P(X_1 X_2\le 2)$. Both variables are independent and each has the probability density function $f(x)=1, 1<x<2$, zero elsewhere. First I would like to assume that the ...
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0answers
64 views

How to prove that $X+Y \mod p$ is indpendent from $X$ if $X$ and $Y$ are independent?

We have a group $\mathbb{Z}_p$ and some random variable $X$ and $Y$ with this domain. We have that $Y$ is chosen uniformly at random, thus each element from $\mathbb{Z}_p$ with probability ...
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1answer
42 views

is this function increasing or decreasing on what intervals?

I have the probability density function for $X$ is $f(x)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\dfrac{-(x-\mu)^2}{2\sigma^2}}$. Let $R=D(X)=\frac{x-\mu}{\sigma}$ and ...
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0answers
31 views

Formulate conditional probability from data

I am working my way through this paper and while I have now managed to obtain all the results contained in it, I am stuck with the rather fundamental problem that I do not understand where equation ...
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1answer
14 views

If $f(x)$ belong to set of probability distributions, then what can be deduced about $\frac{1}{a} f(\frac{1}{a}\cdot x)$

My question is in the context of probability distributions, whose Fourier transforms (characteristic function) almost always exit. If $f(x)$ be some function such that $ \int_{-\infty}^\infty f(x) \, ...
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1answer
158 views

Listing the values of particular probability distributions

Three balls are selected at random from a bag containing 2 red , 3 green , and 4 blue balls. Define the random variables R = the number of red balls drawn, and G = the number of green balls drawn. ...
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1answer
127 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
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1answer
25 views

calculating exponential distributions for products going bad

half of our products (follows an exponential distribution) gone bad during a a week. calculate how long does it take for 1/3 of the products to go bad? my answer (I put this in calculator and get no ...
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2answers
88 views

Is zero random variable a continuous random variable?

In my probability textbook I got stuck on the following problem: cosntruct a positive (it means $\geq 0$) real function $f(x)$ (not necessarily continuous, of course) such that it is constantly $0$ ...
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1answer
26 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...
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1answer
89 views

MLE of $n(\theta,a\theta)$ family

Question: A special case of a normal family is one in which the mean and variance are related, the $n(\theta,a\theta)$ family. If we are interested in testing this relationship, regardless of the ...
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1answer
28 views

The probability that a device doesn't work during specific time interval

Assume both the time to failure and time to repair are exponentially distributed. The failure rate is $\lambda$ and the repair rate is $\mu$. The repair starts immediately after the failure occurs. We ...
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1answer
69 views

Probabilities and Conditional Expectation

Good day! Please check my answers. Here is the problem: Let $ X,Y, Z$ have joint pdf $f(x,y,z) = \frac23 (x+y+z), 0<x<1, 0<y<1,0<z<1,$ zero elsewhere. Find the marginal probability ...
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2answers
65 views

Should I avoid distribution functions in probability?

I'm reading Erhan Çınlar's book on Probability and Stochastics, and in Chapter 2, he says that distributions are used extensively in elementary probability theory in order to avoid measures. And ...
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1answer
40 views

What is application of gamma distribution on pure math or probability theory?

What is application of gamma distribution on pure math or probability theory? i saw it on several probability textbook as a definition, but it seems to me mathematician couldn't derived it if it is ...
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220 views

calculating the expected value of random variable, which is net income

there is 1000 lots in the lottery. you can win 1 unit of 100£, 10 units of 50£ and 15 units of 20£. One lot costs 1£. calculate the expected value of random variable, which is net income. here is my ...
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1answer
144 views

What could be the mathematical model behind “beginner's luck” (followed by losses) in gambling?

I recall a documentary in which a slot machine had trial runs and at first, the desired "bingo" outcome came out more often, but later waned into losses. A scientist plotted the graph, a discrete ...
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1answer
25 views

Is truncating a discrete probability mass function possible?

I have random variable X, and probability distribution: $P[X = A] = .4$ $P[X = B] = .3$ $P[X = C] = .2$ $P[X = D] = .1$ I want to create a conditional probability with event F. Where F is the ...
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71 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
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561 views

Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2 $. Picking the symmetric case with $0$ mean ...
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1answer
188 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
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39 views

Uniform Convergence and exponential inequality to demonstrate Stirling's formula

If $0<R\leq\sqrt n$ and $Z_{k} \sim exp(1)$ are independent and identically distributed as prove that $$P\left[\left|\dfrac{Z_{1}+Z_{2}+\cdots +Z_{n} -n}{\sqrt n }\right|\leq R\right] \geq 1- ...
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1answer
651 views

A cumulative distribution function is discontinuous at some points. Then Random Variable X is discrete?

Here is the distribution So, is the random variable X discrete or continuous? Note: first condition is if x<0 Edit: I have never come across this kind of distributions before. Extensive ...
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1answer
64 views

When computing the CDF from a PDF, why is the integral bound a different variable? $F(x) =\int_{-\infty}^x f(t)\,dt$

Right now, I know the variable in the integral must be x, otherwise, the final result does not match the published CDFs of popular distributions. But I don't know why conceptually this is. If the CDF ...
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1answer
37 views

How to use Chi-Square for a rolling die

I'm watching this video. The guy said that this kind of test allows us to check whether the experiment occurs by chance or something wrong with the way the experiment is being done. I have done very ...
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1answer
111 views

The smallest integer $n$ for a Poisson distribution

Along a stretch of motorway, breakdowns require the summoning of the breakdown services occur with a frequency of 2.4 per day, on average. Assume the breakdowns occur randomly and that they follow ...