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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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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 of 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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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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Exponential Distribution and Markov Chains [closed]

A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1,1.5 and 3 years. Formulate a ...
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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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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 ...
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Let $T$ be an exponential random variable with parameter $\theta$. For $t \gt0$, compute $\Bbb{E}(T|T\le t)$

Let $T$ be an exponential random variable with parameter $\theta$. For $t \gt0$, compute $\Bbb{E}(T|T\le t)$. My work: First $$P(T\le s|T\le t)=\frac{\int_0^s\theta e^{-\theta x}dx}{\int_0^t\theta e^{...
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$X_1, X_2, …, X_n \sim Exp(\lambda)$, what's the joint distribution of $X_1, X_1+X_2, …, X_1+X_2+…X_n$ and is it a uniform ordered distribution?

To elaborate on the title, here is the entire problem: Let $X_1, X_2, ..., X_n \thicksim Exp(\lambda)$ be an independent sample. What's the joint distribution of the sequence of $X_1, X_1 + X_2, ...,...
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sum of two independent exponential distributions?

The sum of two independent exponential distributions \begin{aligned}{Z=\omega_{1}X_{1}+\omega_{2}X_{2}}\end{aligned} is as follows. \begin{aligned}f_{Z}(z)=&\int _{0}^{z}\omega_{1}\omega_{2}f_{X_{...
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Finite Moments of Vector in Exponential Family

I am studying some notes on exponential families and there is a section on the computation of moments. The exponential family has the form $$\exp(\sum_{j = 1}^k \phi_j B_j(x) + C(x) - D(\phi))$$ I ...
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How to find the MLE of these parameters given distribution?

Let $X$ and $Y$ be independent exponential random variables, with $$f(x\mid\lambda)=\frac{1}{\lambda}\exp{\left(-\frac{x}{\lambda}\right)},\,x>0\,, \qquad f(y\mid\mu)=\frac{1}{\mu}\exp{\left(-\...
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Confused about how to show independent random variable $Y$ has the Poisson distribution with parameter $t\lambda$

Assume there are N independent exponential random variable $(X_1, X_2,..., X_N)$ with parameter $\lambda$. Fix a real number $t > 0$. Let Y be the largest $N$ so that $X_1 + X_2 + \ldots + X_N \...
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Exponential Test

Given a collection of data points, without any other information I want to test for an exponential distribution. In a different case, I tested for normality using an Accord implementation of the ...
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Poisson with Exponentially Distributed Parameter

I am doing a review for the test, and I have found myself really struggling with the following question: Prove using generating functions that if $$V \sim \mathrm{Poi}(\theta),~\theta \sim \mathrm{...
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Let $x_1=\exp(\lambda_1),x_2=\exp(\lambda_2)$, $x_1, x_2$ are independent show $P(x_1<x_2)=\frac{\lambda_1}{\lambda_1+\lambda_2}$ [closed]

Let $x_1=\exp(\lambda_1),x_2=\exp(\lambda_2)$, and $x_1, x_2$ are independent random variables. Show that $P(x_1<x_2)=\dfrac{\lambda_1}{\lambda_1+\lambda_2}$
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conditional expectation of gamma distribution with alpha = 1 [closed]

$X_i$ are exponential(\lamda) distribution and identically independent distribution. Y = $\sum_{i=1}^n$$X_i$ $X_i$ is an unbaised estimator of \lamda. Y is a sufficient estiamtor of \lamda. solve ...
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PDF of the time at which the second raindrops hits

Raindrops hit you at a rate or 1 raindrop per second, what is the PDF of the time at which the second raindrop hits you? Clearly, we have to use exponential random variable. Also we are asked to use ...
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Difference of two exponential distribution

Let $X\sim Exp(\lambda_1)$ and $Y\sim Exp(\lambda_2)$, with $Z = X - Y$. I am trying to find the pdf of Z, i.e. $f_Z(z)$. Here is what I have got: \begin{align*} f_Z(z) &= \int_0^{z}...
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Sum of 2 exponential distribution with different parameters

Let $X\sim Exp(\lambda_1)$ and $Y\sim Exp(\lambda_2)$, I am trying to find the distribution of $Z = X+Y$. I understand that $f_z(z)=\frac{\lambda_1 \lambda_2}{\lambda_2-\lambda_1} \left(\exp[-\...
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Probability minimum is “reached”

Let $(X_i)_{i=1,\dots,n}$ be a finite sequence of random variables such that $X_i\sim\mathcal{E}(\lambda_i).$ We can prove that $Y:=\min_{1\le i\le n}X_i\sim\mathcal{E}(\lambda=\sum_{i=1}^n \lambda_i)...
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If $X\thicksim \text{Exp}(\lambda)$ and $Y\thicksim\text{Geom}(p)$ are independent, find $\mathbb P(\lfloor X\rfloor=Y)$

Suppose that $X$ and $Y$ are independent, $X\thicksim \text{Exp}(\lambda)$ and $Y\thicksim\text{Geom}(p).$ Let $\lfloor x\rfloor$ be the floor function (largest interger which is at most $x$). Find $\...
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Probability of Ruin at the first claim

The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution. Assume the initial surplus is $U$, and the ...
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39 views

Probability that lightbulb stops working in odd year

I have a lightbulb that has an exponential lifetime distribution with mean $\mu$ months. So if I construct a pdf $f(x)$ and cdf $F(x)$ with parameter $\lambda$, and since $E[X] = \frac{1}{\lambda}$, \...
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Find the p value following the exponential distribution $\mu=3$

I want to find the $p$-value (manually) of the following Hypothesis testing. $$H_0:\mu\leq 3 \quad \text{vs} \quad H_1:\mu >3$$ The main thing I know is that $$P(\mathrm{Re}\,j \mid \mu \leq 3)=...
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Proof of $P(X<Y)$

Assume that $X$ is $Exp(\lambda)$ distributed and $Y$ is $Exp(\mu)$, and they are independent. I want to know how I can calculate $P(X<Y)$. I don't understand why $$ P(X < Y)=\int_{-\infty}^\...
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29 views

Derivation of the expectation of exponential random variable

I am following some course notes for the expectation of an exponential random variable, $X \sim \text{Expon}(\mu)$. I believe this is a correct derivation: $$ \begin{align} \mathbb{E}[X] &= \int_{...