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

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Distribution of random variables (normal and standard normal)

Suppose that $X_i \sim N(\mu, \sigma^2)$ for $i = 1, \ldots, n$ and that $Z_i \sim N(0,1)$ where all of the random variables are independent. Denote $s^2_Z$ as the sample variance of $Z_1 , \ldots, ...
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3answers
63 views

Hello expected output (probability question)

I am working on a probability problem I tried finding the total net productivity days based on the amount of machines the factory has, so if there was 1 machine, there will be 29 days * 1 machine = ...
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0answers
40 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of (i) ...
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25 views

Selection of Distribution model

An expressed parcel delivery company offers a First Class service for which it is promised that 80% of all parcels are delivered within 24 hours of dispatch. It is suspected that the true successful ...
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72 views

Expected no of balls to select before a certain type of ball comes

There are w white balls and r red balls in a box, to find the expected no of balls to pick before we get a red ball? $$\qquad$$ What I have tried is, Let $ X_k $ denote that k no of white balls have ...
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58 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
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33 views

Conditional distribution of two binomials which both depend on a third

I have a question that I'm having some trouble with, but which I believe might have a fairly straightforward answer. I'd really appreciate it if someone could help point me in the right direction! ...
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1answer
96 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
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2answers
69 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
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1answer
54 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, $f(\...
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1answer
39 views

Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
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1answer
97 views

Bounds-negative binomial distribution

Suppose $Y=\sum_{i=1}^{n} X_{i}$ where each $X_{i}$ is an independently and identically distributed geometric random variable with success parameter $p$, so that $Y$ has a negative binomial ...
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36 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
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2answers
32 views

Find the distribution function of bivariate distribution

Find the distribution function of $$f_{X,Y}(x,y)=\begin{cases} e^{-y}, & \text{if $0< x<y < \infty$} \\ 0, & \text{ otherwise} \end{cases}$$ Trial : According to my calculation $$...
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1answer
48 views

Joint density calculation

Let $X$ have a (standard) normal distribution; with zero mean and unit variance. Let $Y=WX$ where $\mathsf P(W=1) = \mathsf P(W=-1) = \tfrac{1}{2}$. What are the joint and conditional probability ...
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74 views

Probability that the proportion of a shorter segment with relation to the longer one is less than $\dfrac{1}{4}$

The problem is as follows. We randomly pick a point on a segment line of lenght L. What is the probability that the quotient of the shorter segment with relation to the longer one is less than $\...
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2answers
183 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
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1answer
52 views

Probability Joint Density Question [closed]

Suppose $(X, Y )$ is uniformly distributed over the set $\{(x, y) : 0 < y + x < 2, 0 < x < 2\}$. Find the joint density of $(X,Y)$ and marginal density of $F_Y(y)$. I am having a tough ...
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65 views

Exponential distribution of random variable [closed]

Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with ...
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3answers
69 views

Confused by (cumulative) distribution function question…

$P(0<=X<1)$ if $X$ is a random variable having a distribution function: $F(x)=$ {($0, x<0$), ($1/3, 0<=x<1$), ($2/3, 1<=x<2$), ($1, x>=2)$} (hope that makes sense) But if $x$ ...
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1answer
88 views

simplify the division of popular probability density function

This is my first question in Mathematics on Stack Exchange. Please forgive that this is a none sense question... Question I'd like to know a simple form of the division of popular probability ...
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1answer
71 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
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230 views

CDF of sum of $3$ independent discrete uniform random variables on $\{1,2,\dots,n\}$

What is an approximate closed formula for this probability, with a derivation: $p(k,n)$ is the probability, that among $n$ PC disks and $k$ errors in sum on them, there will be at least $1$ disk ...
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1answer
60 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: $Y=10(2+\mu+\...
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51 views

Prove that a function is decreasing

Let $\left(\,c_m\,\right)_{m \in \mathbb{N}}$ be some coefficients which are all positive natural, $c_0=1$, and they are increasing in $m$. Define $$ f(y) = \frac{\sum\limits_{m=0} c_m \, \, ( y \...
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61 views

Computer Component with Gamma Distribution? [closed]

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...
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2answers
48 views

Two company and probability example?

I ran into a problem that seems strange to me. Two companies A,B produce a device that with probability $0.05$ and $0.01$ are broken. if we buy two devices produced by one company with equal ...
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198 views

Stochastic dominance of Binomial and Poission

In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove. Namely, I am looking for a proof of ...
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1answer
39 views

Distribution of P[Y=n] = P[n-1<X<n] for X exponentially distributed

From an assignment, we have "Let X be an exponentially distributed random variable with probability density function. $f(x) = λe^{−λx}$, for $x > 0$" I've worked out that for $P[Y=n] = P[n-1 < ...
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Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
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105 views

What distribution models number of trials needed for given number of successes and success rate?

Case scenario: a retro-virus infects a healthy cell. The virus programs the cell to brew little viruses, at a rate of 0.5 per-sec, until finally the cell bursts when the number of virus inside it is 5....
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60 views

Simulating r.v.'s from a joint density by rejection sampling in R. Continued

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
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Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let $g_i(x)=f_1*...
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1answer
88 views

Is the support of the Gaussian finite or infinite?

Considering that as $x \to \pm \infty$ ; $e^{-\frac{x^2}{2}} \to 0$, is the support finite or infinite? A simple enough question, but enough to make me scratch my head. I feel that it's almost a ...
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174 views

Probability density function for distance between two points.

Two points are chosen randomly inside a circle (and even on the circumference) with radius $r$ What is the probability density function of the distance between the points? I would be very grateful.
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24 views

Differentiability of CDF at 0

This might seem to be a very trivial question but anyway here we go: I'm currently reading the paper "On the Value of a Random Minimum Spanning Tree Problem" by Frieze (1984) and I'm stuck on the ...
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45 views

Finding density and distribution functions [closed]

I have been trying to understand probability by attempting past paper question and I have been stuck on this question all day and night. I am not quite sure how to go about finding the functions ...
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1answer
164 views

A problem on probability concerning distributions of particles

I saw this problem in An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). I am quite intrigued by the problem. Here it is: Suppose that a particle is equally likely ...
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14 views

Looking for a distribution to describe 2D “lines”

I have a 2D surface endowed with line segments, i.e. the function contains sparsely distributed segments in which there is high correlation between adjacent points in some direction, and the rest of ...
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1answer
91 views

Binomial? probability of two producing defective articles machines.

Suppose that machine A produces (on a daily basis) twice the articles that machine B produces. However, 4/100 of the articles produced by machine A are defective while 2/100 of the articles produced ...
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77 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
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63 views

Constructing a piecewise probability density function

Here I'm trying to construct a probability density function in the form $$f(t) = \begin{cases} at, & t \in [0, 5) \\ b\sqrt{t}, & t \in [5, 20]\text{.} \end{cases}$$ Of course, $$\int\limits_{...
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Probability distribution of request handling

I have values representing time taken to execute one request on server. Could somebody advise what type of distribution it is? I think that normal distribution but I am not really sure about it. ...
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72 views

Dependent Bernoulli trials confidence interval

I would like to know if there is a way to build a confidence interval, for a random variable which has a Bernoulli distribution, based on its history. I mean if the order of its states is 11100 (i.e. ...
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320 views

how to determine transient and recurrent state from transition matrix

I wonder how can I determine the transient and recurrent state from transition matrix ? I mean if I have 10 states It would be very hard to draw diagram for them so how to analyse the matrix? For ...
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For which values of $ a $ can this function be a valid distribution function?

Let $ F(x) = a(x+1)^2(u(x+1)-u(x-1))+u(x+1) $ . For which values of $ a $ can this function be a valid distribution function? I couldn't solve this question. Because for $x \geq 1 \implies F(x) = 1$...
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31 views

CDF of two variable

I would like to calculate the CDF of sum of two random variable in a unit square I realize that everywhere says if X+Y=z and then if z is between 0 and 1 then probability is equal to something and if ...
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20 views

probability of joint PDF

I found $k = 4$ and yes, the are independent. But for the last one I know how to find the probability if they are like $x$ from $0$ to a number and $y$ from $0$ to a number so the limit of double ...
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52 views

Pdf of variable as combination of two random variables with exponential distribution

If $X$ and $Y$ are independent and exponentially distributed, which is the pdf of $Z$? Where $Z$ is given by \begin{equation} Z = \frac{X}{1+Y} \end{equation} I read answer to this post: $X,Y$ are ...
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21 views

Representation of a non-standard normal variable squared

I have come across a representation of a non-standard normal distributed variable square. It is clear for me that assuming $Z_j \approx N\left ( \theta_j, \frac{\sigma^2}{n} \right )$ we can write ...