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

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111 views

Probability in DNA segmentation

I have formulated these questions ss part of a research in medical science (DNA segmentation): A series of $M$ identical balls is arranged on a line. A partition is formed by placing a stick to ...
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1answer
32 views

covariance of a transformed Gaussian matrix

Suppose $X \in R^{m \times n}$ is a matrix with each entry being an i.i.d Gaussian random variable with 0 mean and unit variance. Suppose $A,B$ are known matrices of appropriate dimensions. What is ...
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1answer
315 views

Analogous of Markov's inequality for the lower bound

Consider a positive random variable $X$ and call $E[X]$ its expectation. For any positive $a \in \mathbb{R}$, an upper bound for the probability of $P(X>a)$ is provided by the Markov's Inequality, $...
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31 views

solving for an unknown within expections with known distribution

I have this equation to solve for $x$. $$ E\left[ e^{xa_i}a_i\right]=c E\left[ e^{xa_i}\right] $$ where $a_i$ is a RV and it's logarithm is distributed as $$\ln a_i \sim N (-\frac{\sigma^2}{2}, \...
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31 views

Probability question on 5 components operation after a given time

My attempt: $\lambda=1/2.5=0.4$ Since $P(T\geq t)=({1-e^{-\lambda t}})$ $P(T\geq3)=({1-e^{-0.4(3)}})^5$ However my book says: $P(T\geq3)=({e^{-0.4(3)}})^5$ Why is this? How did the book do it....
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2answers
33 views

Evaluating an expectation of the form $\mathbb{E}f(X)$

Suppose I have scalar random variables $$X_1 + \cdots + X_n,$$ defined on some probability space and let $S_n$ be their sum. How can I prove the identity $$\mathbb{E}|S_n|^2=\sum_i^n\sum_j^n\mathbb{...
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1answer
30 views

What distribution $\frac{1}{d} \sum_{i=1}^{d}X_i $ will follow if $X_i$ is a bernoulli random variable?

I am a newbie on probability and statistics. On the course of studying binomial distribution, I was curious about the distribution of 'sample mean' of binomial distribution. Formally, the ...
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0answers
150 views

Probability of drawing balls with different colours

In a jar of balls with different colours, can we find a general probability distribution of having n distinct colours with N number of balls grabbed ? Assuming that any colours have an equal chance to ...
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2answers
2k views

Applying the Negative Binomial Distribution to problems.

A family decides to have children until it has three children of the same gender. Assuming $P(B) = P(G) = 0.5$, what is the pmf of $X=$ the number of children in the family? This problem is in the ...
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1answer
38 views

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
76 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 $\int^{\infty}_{...
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1answer
91 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
150 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
113 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) = \frac{x^{a-1}e^{-...
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0answers
247 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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128 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 exp(-|x|...
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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
58 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
71 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
97 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 ($x_i=...
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2answers
114 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
157 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
177 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 $f(x,y)=21x^2y^3, 0<x<y<1$...
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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: $$\begin{align*}P(...
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1answer
28 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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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 $\frac{1}{p}...
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1answer
44 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 $-\infty<x<\infty,-\infty<\mu&...
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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 (1)...
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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
162 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
128 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 food,...
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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
92 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
95 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
70 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
67 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 again,...
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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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2answers
228 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
149 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 ...