Questions tagged [exponential-distribution]

To be used for questions on using, finding, or otherwise relating to Exponential Distributions.

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11 views

Joint distribution and covariance of poisson process and waiting time

Hi I am having a trouble solving for this problem where I have to find 1) Joint distribution of $W_{1}$, $W_{r}$ for $r\geq2$. 2) $\operatorname{Cov}(W_{1},W_{r})$ for $r\geq2$. [Notation ...
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An exponential distribution that represents time between events

You're responsible for maintaining four ATMs (E,W,N, and S). The time between failures for each ATM is exponentially distributed with mean time between failures 6 hours, 5 hours, 8 hours, and 8 hours, ...
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Density function of product of random variables.

I've got the next question: Let $X,Y$ independent random variables such that: $$ X \sim Unif(-1,0) \quad \mbox{ and } \quad Y \sim Unif(0,1).$$ Find the density function of $Z=XY$ using the density ...
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Exponential random variable problem

I have an exercise that give me a lot of problems during the resolution and I hope in your help... The statement following... We have an exponential random variable with variance $\sigma ^{2}$, with ...
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1answer
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Poisson or exponential distribution

I have to complete the question for a homework task and I'm confused if it is a Poisson or exponential distribution. any insight would be appreciated. Voters arrive at a polling booth in a town ...
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When given $X = −\frac{\ln(1 − U)}{\lambda}$, why is $X \sim \mathrm{Exp}(\lambda)$ and not $\mathrm{Exp}(-\lambda)?$

When given $X = \frac{−\ln(1 − U)}{\lambda}$, why is the distribution of $X \sim \mathrm{Exp}(\lambda)$ and not $\mathrm{Exp}(-\lambda)?$ I solved for $X$ to get: $-X=\cfrac{\ln(1-U)}{\lambda}$ $-\...
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26 views

Probability: Sum of exponential distributions

Suppose you have three random variables $X, Y, Z,$ where $X\sim \text{Exponential}(2)$, $Y\sim \text{Exponential}(1)$, $Z\sim \text{Exponential}(2.5)$. Let the random variable $D$ be given by $D= 3X + ...
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the confidence intervals of an exponential distribution [closed]

Consider the random sample $X_1 \cdots X_n$ from a distribution with pdf $$f(x; \theta) = \dfrac {x}{2\theta^2}$$ if $0<x<2 \theta$. The most likelihood estimator of theta is given by: $\theta_{...
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1answer
30 views

How to calculate the joint probability: $\Pr \left( \tfrac{g_1}{g_3} \geq \theta_1, \tfrac{g_2}{g_3} \geq \theta_2, g_3 > \theta_3 \right)$?

Question: How to calculate the following? $$\Pr \left( \dfrac{g_1}{g_3} \geq \theta_1, \dfrac{g_2}{g_3} \geq \theta_2, g_3 > \theta_3 \right),$$ where $g_i, i \in \{1, 2, 3\}$ is an exponentially ...
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Stochastic Process problem. Poisson procces

Couldn't solve one exercise. Exercise is as follows: The Bank employs $10$ operators. The service time for each is exponentially distributed, with the average $i$-th operator serving $i$ clients per ...
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One Poisson process arrives before the other: explanation of method

Let $𝑋_𝑡$ and $Y_t$ be two independent Poisson Process with rate parameters $\lambda_1$ and $\lambda_2$ respectively, measuring the number of customers arriving in stores 1 and 2. What is the ...
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42 views

Finding the probability of the Exp($\lambda$) distributed minimum of X (given a pdf)

I am trying to find the following probability of the $\min(x)$ of a exponential distribution: $$P\left(\frac1{\lambda} \ge \frac{n\min(x)}{\ln(5)}\right) = ?$$ I have the following pdf of $\min(x)$ ...
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1answer
35 views

Finding $P(X+Z>Y)$ where $X,Y,Z$ are exponential random variables

Let $X$,$Y$,$Z$ be independent random variables with exponential distribution of parameter $\lambda$, then $X,Y,Z$ ~ $\xi(\lambda)$. The task is to calculate $P(X+Z>Y)$. Comment: In previous ...
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Mean square error of MLE

$\textbf{X}=(X_1,...,X_n)$ is a random sample from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \...
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Lawfulness of $F_{X|Z}(x,z)=\mathbb{P}(X\leq x|Z\leq z)$

Let $X \perp Y$ two random variables i.i.d. with exponential distribution of rate $\lambda$. Find: 1) the distribution of $W=X+Y$. $\rightarrow W\sim \Gamma(2,\lambda)$ 2) the distribution of $Z=\...
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Determining an unbiased estimator

Say we have a shifted exponential distribution with common density $$f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \end{matrix}\right.$$ We have $\theta$ ...
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$\min(S^3,T)$ with $S \perp T$

Let $S$ and $T$ two random variables with exponential distribution of rate $\lambda$ and density $f(u)=\lambda e^{-\lambda u},u>0$. Find the density of: 1) $X=|S-T|$. $\rightarrow X\sim Exp(\...
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Name for a simple generalization of an exponentially distributed random variable

An exponentially distributed random variable $X$ with mean $\mu$ has a simple survival function: $S(k) := \Pr (X>k) = \exp(-k/\mu)$ for $k$ and $\mu$ non-negative. Consider generalizing to $S(k) :=...
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2answers
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How to calculate the probability density for $W=\frac{Y}{\max(X,Y)}$?

Assume that $X,Y$ are two independent exponential random variables with mean $\lambda$. How can I calculate the probability density function of $W=\frac{Y}{\max(X,Y)}$? I know how to calculate the ...
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Is my average chance of death for a cohort correct

I am trying to model a human population. My simulation calculates at 1 day intervals. I want to track my population counts in cohorts of varying sizes (age 0, 1-14, 15-29, etc). The chance of death ...
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22 views

Moment Generating Function of an Exponential variable

I know that when if we have an exponential random variable with parameter $\lambda$, the moment generating function is $\frac{\lambda}{\lambda-t}$ when $t < \lambda$, but what can I say about the ...
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Waiting time to the 1st event less than t and waiting time to 2nd event greater than t, where waiting times have different rate parameters

Consider a process in which the waiting time (call this X) until the first event is distributed as an exponential random variable with rate parameter lambda_1. Conditional on (X = x), the waiting time ...
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Prove that $n \cdot\min\{T_1,…,T_n\}$ isn't allowable as an estimator of $\mu$

Let's suppose we have some electronic device which duration follows an Exponential distribution of unknown mean $\mu$. Some research team wants to estimate $\mu$ and uses a sample of $n$ devices to do ...
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Problem calculating the characteristic function of the exponential distribution

I was trying to calculate the characteristic function of the exponential distribution $$\varphi(t) = \mathbb E[e^{itX}] = \int_{-\infty}^\infty e^{itx} \lambda e^{-\lambda x} \cdot 1_{[0,\infty)}(x) \,...
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Exponential distribution and the age of the human civilization

So, thinking another day about how young our human civilization is, compared to the potential time it can last for - for example, 5 billion years until the Earth dies, I thought that maybe it is not a ...
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1answer
27 views

Which of this two estimators of $\mu$ is better (Exponential distribution)?

The problem goes like this: "Suppose we have some electronic device which duration follows an Exponential distribution of an unknown mean $\mu$. We want to estimate $\mu$ and two teams will take care ...
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1answer
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Why the process $n \mapsto e^{-aT_n}f(X_{T_n})$ is a super martingale.

Let $M := (e^{-aT_n}f(X_{T_n}); \mathcal{F}_n)$, where $a$ is constant, $\mathcal{F}_n$ is appropriately defined filtration and $T_n$ is $n$th jump of an independent (independent of $X_t$) Poisson ...
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55 views

Conditional and joint distribution of the sum of exponential RVs

Let $X_1,X_2,...,X_n$ be i.i.d. $Exp(\lambda)$ random variables and $Y_k =\sum^{k}_{i=1}X_i$, $k = 1,2,...,n$. a) Find the joint PDF of $Y_1,...,Y_n$. b) Find the conditional PDF of $Y_k$ ...
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1 minute of call costs 1 currency. Let's assume that duration of call has exponential distribution with a parameter $\lambda=1$

1 minute of call costs 1 currency. Let's assume that duration of call has exponential distribution with a parameter $\lambda=1$, which implies that call lasts for an avarage of 1 minute. How much on ...
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206 views

Maximum Value given Second Smallest Value [Exponential(1) Distribution]

Let $X_1, \ldots ,X_n$ be an independent and identically distributed sequence of Exponential(1) random variables, where $n \geq 3$. Find the conditional probability density function for the maximum $M ...
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2-parameter exponential distribution of the packet source [duplicate]

I was doing my homework and the problem came up. Task: We have the maximum value (σ [packets/s]) and mean value (λ [packets/s]) of traffic generated by the packet source. The packet source is ...
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Measurement of the traffic generated by the packet source

I'm working on resolving a statistical problem and I came across some difficulties. The content of the task: A measurement of the traffic generated by the packet source indicates that the average ...
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Using [−∞,0] instead of [0,∞] limit for a convolution difference of independent exponential variables

Let $X_1∼\exp(λ)$ and $X_2∼\exp(λ)$ be two independent exponentially distributed random variables. Find the pdf of $Y = X_1−X_2$ through convolution. My approach: Integrating the product of their ...
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Is this an exponential distribution?

I have a probability density function $f(x) = k \cdot 3e^{-3x}$, with $k\ne 0$ constant. I saw someone saying this is the exponential distribution with $\lambda = 3$. However, isn't the exponencial ...
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Birth Rate Question in Stochastic process

Della has four tasks that need to be completed, which must be performed in order. The times taken to perform each task are independent exponentially distributed random variables. He expects that on ...
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Probability about lifetime of 100 bulbs (exponential distribution)

I have some doubts about the following problem: I have 100 bulbs with a lifetime represented by an exponential distribution, with an expected value of 1000 hours. Find the probability that, at least ...
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Tennis Court - Stochastic Processes Problem

There are two tennis courts. Pairs of players arrive at rate 3 per hour and play for an exponentially distributed amount of time with mean one hour. If there are already two pairs of players waiting, ...
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To derive -log(Uniform)~exp(1)

Q. Let X ~ Uniform(0,1) and Y= -log(X). Find E(Y), Var(Y) A. U ~ Uniform (0,1) 1-U ~ Uniform(0,1) -log(1-U) ~exp(1) -log(U) ~ exp(1) E(Y) = V(Y) = 1 I can't understand the -log(U) ~ exp(1) and E(...
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Formula to find mean & median for exponential distribution.

I need a bit of help from you guys. I have a function like this : $F(X)=\frac{95}{1-X}$ (in python) def f(x): return math.floor(95 / (1 - x)) Where X is a ...
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Distribution of minimum of two independent exponential distributions

Suppose $\lambda, \mu > 0$ and $X \sim \mathsf{Exp}(\lambda)$ and $Y \sim \mathsf{Exp}(\mu).$ We were asked to find the distribution of $\min(X,Y).$ Let $\ell$ be in R. $P(\min(X,Y)< \ell)= P(X &...
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Properties of a Poisson Distribution

So if we were to say $X(t)$ is a Poisson process with rate $λ$, I'm trying to understand this idea: If we fix $n$ then we would expect the time between the $n^{th}$ arrival and the $(n+1)^{th}$ ...
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Legendre transform in classical mechanics and entropy maximization

I am trying to get some understanding of convex optimization, and in particular of why the Legendre transform appears in certain optimization problems. I am particularly interested in two examples, ...
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Exponential distribution and Markov chain problem

A, B and C student arrive at the beginning of a professor's office time. The duration of time they will stay is exponentially distributed with means of 1, 1/2,and 1/3 hour. I want to find the ...
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Probability of correctly classified phone calls that have an exponential distribution

Background: A game company produces game A and B. The company will receive phone complaints from the gamers every day. 60% of complaints are from people who play A and 40% are from people who play ...
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What is the relation of this result to thinning? Construct a second probabilistic proof using thinning of a Poisson process.

If $\{E_{n}\}$ are idd exponentially distributed with parameter $\alpha$, $N$ is geometric with $$P[N=n]=pq^{n-1}, \quad n \geq 1,$$ and $N$ is independent of $\{E_{n}\}$. Here I show that $$\sum_{n=1}...
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51 views

how to derive the expected value and variance of this exponential distribution

I'm new to probability and I'm stucking with this problem. Let $X$ be a continuous random variable following a pdf $$p(x|\theta)=\frac{\theta}{2}\exp(-\theta |x-\mu|)$$ Then, how can I calculate the ...
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Definition of hyperexponential distribution

I'm reading lecture note Continuous-Time Markov Chain Models by V. Andasari. Suppose $X_{1}, X_{2}, \cdots, X_{n}$ are independent exponential random variables. If the exponential random variables $...
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Find the probability of $P(t<X_1<X_2)$ where $X_i\sim \exp(\rho_i)$ and independent from each other.

Let $X_i\sim \exp(\rho_i)$ independent from each other. How can I show that the following hold, $P(t<X_1<X_2)=\int_t^\infty P(x<X_2)f_{X1}(x)dx=\frac{\rho_1}{\rho_1+\rho_2}e^{-(\rho_1+\...
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28 views

Exponentially distributed random variable inquality (using memoryless property)

This is my first post :) So given a exponentially distributed random variable $X$, how do I show that for $t,s$ positive integers and $u\in [0,s]$: $$ P(X>t+s-u)=P(X>t)P(X>s-u) $$ I know that ...
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Exponential Likelihood Function

Suppose $X_1, ..., X_n \stackrel{iid}{\sim}$ Exponential(rate = $\lambda$) independent of $Y_1, ..., Y_n \stackrel{iid}{\sim}$ Exponential$(1)$. Define $Z_i \equiv \min\{X_i, Y_i\}$ I want to find ...

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