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Questions tagged [exponential-distribution]

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

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Variance of the range for the exponential distribution

If $n$ random variables are independently distributed with an exponential distribution $f_X(x) =\lambda e^{-\lambda x} (x\geq 0)$, the range $R_n = \max(X_1,\cdots,X_n) - \min(X_1,\cdots,X_n)$ has the ...
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Transforming sum of n exponential distribution to a Poisson distribution

Let $X_1,...,X_n$ be i.i.d exponential random variable with mean $\lambda$ $S=X_1+...+X_n$ So by finding the mgf of S, we get that $S \sim \operatorname{Gamma}(n,\lambda)$ The problem I am stuck ...
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how to derive this property of exponential distribution?

Let $X$ be a real random variable with exponential distribution on a measure space with probability measure $\Bbb P$. Let $\Bbb E$ be the expected value. Then \begin{equation}\Bbb P(X>s+t|X>s)=\...
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20 views

Conditional probability with exponential distribution

Suppose that $X \sim \mathcal{E}(1.3)$ and $Y \sim \mathcal{E}(1.7)$ are two exponential random variables and define $U := \min\{X, Y\}$. How do I calculate following values? $\mathbb{P}[U > 0.32 ...
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Finding a confidence interval for shifted exponential distribution

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\sigma ,\theta)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}, x\gt \theta$$ where $\sigma \gt 0 $ and $\theta \in R$ . a) if ...
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On average, 3 oil tankers arrive at a port each day, The time between arrivals follows an exponential distribution.

(a) Find the probability that the next tanker does not arrive until at least two days from now. (b) Find the probability that seven tankers arrive in one day. for first question we have the ...
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24 views

Bulbs with amnesia

Here is a question for which I am not able to figure out the approach to solving it. Problem statement: Suppose that $n$ light bulbs in a room are switched on at the same instant. The life time of ...
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Geometric distribution of a Yule process multiplied by $e^{-\lambda s}$ converges to exponential distribution with parameter 1

If Y is a Yule process with rate $\lambda$ than I know that the generating function is $$F(x,t) = \frac{xe^{-\lambda t}}{1-x+xe^{-\lambda t}}$$. Now I want to show that $$\lim_{s\to \infty}e^{-\...
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Mean time to sharded data being unavailable in a distributed storage system

How do I model time to data unavailability given the following parameters? Data unavailability means that every machine that hosts a replica of data is down at the same time. All replicas for a ...
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Calculate the MLE of 1/𝜆 from exponential distribution

I have a sample of n > 2 independent and identically distributed random variables Y1, . . . , Yn from an exp(λ) distribution with probability density function. I know that the MLE of λ $$\lambda = \...
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34 views

Pivotal Quantity for the location parameter of a two parameter exponential distribution

Let $X$ be a random variable with probability density function $f(x,\theta, \beta)=\beta e^{-\beta(x-\theta)} \mathbb{1}_{(\theta,\infty)}$ with $\beta>0, \theta \in \mathbb{R}$ (a two parameter ...
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Natural Parameter Space of an Exponential Family is a convex set (Proof)?

A natural parameter family is defined as follows $$p(x|\eta) = h(x) exp(\eta T(x) + A(\eta))$$ where T: sufficient statistics A: log partition function. We want to prove that the natural parameter ...
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22 views

Min/max probability distribution

I am having a very hard time understanding probability problems involving max and min distributions, for example: For $n$ i.i.d. exponential random variables, find $\Bbb P(X_1=\text{min}(X_1,...,X_n))...
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Finding the mean of an exponential distribution that starts at time T

Given an exponential distribution $\lambda e^{-\lambda t}$ for $t \geq 0$, the mean is $1/\lambda$. I'm wondering how one would compute the mean if the process starts at a certain time and we start ...
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59 views

Expected Waiting Time in a Queue with exponential distribution

I am trying to solve the following problem in my exercise sheet towards preparation of competitive exam: A post office has 2 clerks. $A$ enters the post office while 2 other customers, $B$ and $C$, ...
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24 views

Expectation of -log(U)

Let $U$ be a uniform distribution on $[0,1]$ 1) Find the distribution function of $V = -log(U)$ (where log is the natural log) 2) Find $E(V)$ What I got: 1) $F_V(x) = P(V<x) = P(-log(U) < x) ...
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UMVUE of $P(X_1 ≥ t)$ for a two-parameter exponential distribution

I'm attempting to find $(a)$ The UMVUE of $λ$ when $θ$ is known. $(b)$ The UMVUE of $θ$ when $λ$ is known. $(c)$ The UMVUE of $P(X_1 ≥ t)$ for a fixed $t > θ$ when $λ$ is known. I'm new to the ...
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Deriving exponential distribution from arrival time probability.

Consider the following question from "Lectures on Probability Theory and Mathematical Statistics": The probability that a new customer enters a shop during a given minute is approximately 1%, ...
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60 views

Distributions of $U=\max\{X-Y,0\}$ and $V=\max\{X,Y\}-\min\{X,Y\}$ where $X,Y$ exponential

Suppose $X\sim \text{Exp}(\lambda)$ and $Y\sim \text{Exp}(\mu)$ are independent. I've shown that $Z=\min\{X, Y\}$ is independent of the event $\{X<Y\}$. I want to find the following: $P(X=Z)$ ...
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Devising a hypothesis test for machine failure rate

I'm trying to devise a hypothesis test for failure rate data of machines. The gist is that there are some machines in a factory that run all the time. They fail from time to time and are promptly ...
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Computing CDF from an exponential r.v. X and Bernoulli r.v. Z

I need to compute the CDF of Y=ZX and i am struggling to compute this when Y=ZX Information: X is an exponential random variable with parameter 1 Z a Bernouilli random variable taking its values ...
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Asymptotic distribution of MLE of joint exponential distributions

Given $X_1,...,X_n \sim \text{Exp}(\theta)$ and $Y_1,...,Y_n \sim \text{Exp}(\frac{1}{\theta})$, where $\theta>0$ and have the same $\theta$ in both distributions. $X$ and $Y$ are independent. I ...
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Characteristic function evaluated at complex number for exponential distributed random variable

Given: $$f_X(x) = \lambda e^{-\lambda x},\; x\in X$$ Wanted: The corresponding characteristic function $\phi(ju)$. \begin{align} \phi(ju)&=\mathbb{E}(e^{j^2ux})\\ &= \lambda \int^{\infty}...
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How can I find the expected value of this variable?

The initial cost of a machine is 3\$. The life-time, T, of that machine has an exponential distribution with an expected value equal to 3 years. The maker wants to offer a warranty that pays 3\$ if ...
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Joint Exponential Probability Problem

I have a question from A First Course in Probability by Sheldon Ross. Question: Consider two components and three types of shocks. A type 1 shock causes component 1 to fail, a type 2 shock causes ...
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MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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44 views

Probability generating function of exponential distribution

The exponential distribution is given by: $$PDF: \lambda e^{\lambda x}$$ And the formula for probability generating function is given by: $$G(z) = \sum_{x=0}^\infty p(x)z^x$$ where $p(x)$ is a ...
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Sufficiency in the exponential distribution

I am trying to show that given a random sample $\{X_i\}_{i=1}^n$ where $X_i\sim exp(\lambda^{-1})$, the statistic $T(\mathbf{X})=\sum_{i=1}^n X_i$ is sufficient by using only the definition. I have ...
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Find the pdf , distribution function of $X$ and $E[(X-2)^2]$

I 'll be very grateful if you can help me , here is the question : When a person sends an email, the probability that there is an attachment is 0.5. If there is an attachment then the size of the ...
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Exponential distribution MLE with lifetime and frequency table

\begin{array}{c|c} \hline \text{Lifetime (months)} & \text{Observed frequency} \\ \hline 0-2 & 50 \\ \hline 2-4 & 35 \\ \hline 4-6 & 25 \\ \hline 6-8 & 15 \\ \hline 8-10 & 5 ...
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Hypothesis testing for the lifetime of bulbs using subsystems

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean λ. To test the null hypothesis H0 : λ = 1000 against the alternative ...
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Introduction of shape parameters in the formulation of probability distribution

I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the ...
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a constant $k$ of a Weibull Distribution

I am supposed to show the constant of a Weibull Distribution, $k$, is a product of $\alpha$ and $\beta$. I know $\alpha = \frac{1}{\theta}$ ($\theta$ being the probability of a success) for the ...
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Why isn't expected time to leave a system with exponential service times simply the shortest expected service time times two?

This is from exercise 25, chapter 5, in Ross' Introduction to Probability Models, 11th ed. It goes as follows: Customers can be served by any of three servers, where the service times of server $i$ ...
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If $X$ is an exponentially distributed variable with mean $ \lambda$, $Y=−3\ln(X)$ has Gumbel distribution?

Let X be a random variable which follows an exponential distribution with parameter $\lambda$ ($\lambda>0$), find the distribution of the random variable $Y = −3\ln(X)$. So this is my answer for ...
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Expectation of number of broken machines by time t

$n$ machines in total (starting working at the same time), working time of each is i.i.d. $Exp(\lambda)$. How to calculate the expectation of the number of broken machines at time $t$? (If broken, ...
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Probability between two events following exponential distribution

Two independent events A and B follow exponential distribution with parameter L: f(t)=Le^(-Lt) for t>=0. If X is the time where A occurs and Y the time where B occurs, calculate the probability P[X>=2*...
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Coxian distribution: probability first phase smaller than x

I'm trying to solve a question about the coxian distribution. There are two phases, phase 1 and phase 2. Because coxian is not so famous, I will give the definition as in our book: Let $B(x)~\sim~C_2(...
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Probability of falling meteor (Poisson process)

Let's say, that we know, that in some period of year there are really common sightings of meteors with an average of 100 meteors per hour. What's the chance of no meteor seen in 5 minutes? I was ...
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How do we obtain an estimate for $\textbf{P}(X\geq 1)$ where $X\sim\text{Exp}(\lambda)$?

Consider a simple sample $X_{1},X_{2},\ldots,X_{n}$ whose distribution is given by $X\sim Exp(\lambda)$. (a) Determine an estimator for $\lambda$ according to the method of moments. (b) Determine ...
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How to decelerate from velocity v to stop time t over distance d?

I'd be grateful for some help with this problem I am trying to solve. Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
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How to distort a sigmoid / logistic function?

Please help. I need to move an object such that its distance / time profile resembles a sigmoid curve but in a non-symmetrical manner. 1- Let's say that I need to move 800cm in 3 seconds. How do I ...
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Poisson from Exponential

The probability distribution of time between events following a Poisson point process with parameter $\lambda$ is an Exponential Distribution with parameter $\lambda$. The proof from Poisson to ...
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Poisson Process from independent non-identical exponential RVs

I know, I can define a Poisson Process using a sequence of i.i.d. exponential random variables, i.e. let $\tau_1, \tau_2, \tau_3, ... \sim \mathrm{Exp}(\lambda)$, then $T_i = \sum_{j=1}^I \tau_j$ are ...
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Let $X$ be a non-negative random variable, and $A_i$ be the event that $X$ is between $[i - 1, i)$.

Let X be a non-negative random variable, and $A_i$ be the event that $i-1 \leq X < i$. (a) Show that $\sum_{i=1}^{\infty}(i-1)I_{Ai} \leq X < \sum_{i=1}^{\infty}iI_{A_i}$, where $I_{A_i}$ = 1 ...
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Combination of Exponential distributions question (with different probabilities)?

I am currently learning about Poisson processes and I was thinking about the following question. The waiting time for your lunch to be collected by delivery drivers A and B are distributed as $\...
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Intuition behind $E(X-t \; |\; X>t) = E(X)=\frac{1}{\lambda }$ in exponential distribution

Let $X$ be a an exponentially distributed random variable with rate $\lambda $ and let $t$ be a constant. It was shown to me that $$E(X-t \; |\; X>t) = E(X)=\frac{1}{\lambda }$$ However, the ...
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Find $P(X_1+X_2<X_3)$

Given that $X_i\sim Exp(\lambda_i),i\in\mathbb{N}$, find $P(X_1<X_3)$ $P(X_1+X_2<X_3)$ I know that for 1. $P(X_1<X_3)=P\big(X_1=\min\{X_1,X_3\}\big)=\frac{\lambda_1}{\lambda_1+\...
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28 views

Lambda and mean of sum of 2 independent exponential random variable

I have a restaurant with service time being exponentially distributed. Let's say the food-serving time has a mean of 30 minutes, and the checkout time has a mean of 10 minutes. Would $\lambda$ in ...
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Poisson Process with Expected Time (Stochastic Processes)

Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 30 minutes to clean the kitchen. Kelly ...