Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [exponential-distribution]

A distribution describing the time between events in a Poisson process; a continuous analogue of the geometric distribution.

1
vote
0answers
15 views

Passion Distribution [closed]

Suppose the number of tsunami in a season follows a Poisson distribution and the average number of tsunami that hit a region is 5 in every tsunami season A tsunami season lasts for 3 months. In the ...
0
votes
0answers
7 views

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 ...
0
votes
0answers
12 views

Parameter estimation of exponentialy distributed variable with bounded observation

I have data as below. I believe it is exponentialy distributed. But I could not observe the whole data, since I have limited time. The below table shows the monthly frequency. How can I estimate the ...
0
votes
0answers
15 views

Exponential distribution and Poisson process [duplicate]

Could someone please explain to me what is exponential distribution and poisson process mean? How they are different and the relationship between them? [In simple terms]. Thanks
1
vote
0answers
25 views

Exercise about exponential distribution

My textbook has this exercise, in the section regarding exponential distribution: Given an arrival process with $\lambda = 8.0$, what is the probability that an arrival occurs in the first $t = 7$ ...
0
votes
0answers
35 views

How to calculate the Product between Gaussian and exponential distribution in Matlab?

I require to calculate the Product of Gaussian distributed and exponential distribution. My work clamp force = (force of the wheel * friction coefficient) where the force of the wheel is Gaussian ...
1
vote
2answers
57 views

Estimator for $\frac{1}{\lambda}$ using $\min_i X_i$ when $X_i$ are i.i.d $\mathsf{Exp}(\lambda)$

Let $X_1,\ldots,X_n$ be i.i.d. $\mathsf{Exp}(\lambda)$ random variables, where $\lambda$ is unknown. Consider $f_{\min}(x) = \min_{i}(X_i)=$ $ n \lambda $ Exp$(n\lambda x)$. I am told that $\hat \...
0
votes
0answers
50 views

How do I calculate a Bayesian Posterior Distribution from an Exponential Prior and Sample Data

I have a dataset where each observation is a length of time (e.g. 50 days, 70 days, 105 days) and I am trying to utilize Bayesian statistics to calculate a posterior distribution in light of new data. ...
1
vote
1answer
44 views

Proof for simulation of NHPP by thinning

Background: I'm trying to show equivalency between the density function for a non-homogenous exponential process (NHEP?), (i.e. the arrival times of events generated by a non-homogenous Poisson ...
2
votes
0answers
20 views

Trouble understanding derivation of probability for continuous time markov chain

I'm working on exercise 6.10 from "Introduction to probability models" by Sheldon M. Ross. There's an expression for the probability $P_{00}(t)$ that I don't understand. Here's the relevant ...
0
votes
1answer
56 views

Minimal sufficient statistic for location exponential family

Let $X_1,\dots,X_n$ iid with pdf $$f(x|\theta)=e^{-(x-\theta)},\,\,\,\theta<x<\infty,\,\,\,-\infty<\theta<\infty.$$ Part (b) of Problem 6.9 in Casella and Berger asks to find a minimal ...
0
votes
1answer
107 views

Expectation of standard exponential squared given sum of two standard exponentials

So I have been working on this question for a while and made some progress , but I run into a problem about the normalizing constant. The question is, for $X$ and $Y$ i.i.d. standard exponential, find ...
0
votes
0answers
18 views

Explanation for Cumulative Distributive Function example

I'd like to ask for clarification of the following example in my textbook. Example: Suppose events are occurring at random with average rate $\lambda$ per unit of time. What is the probability ...
0
votes
0answers
54 views

Confidence interval with only one observation

How could one create a 95% lower confidence interval for the expectation of a exponentially distributed r.v. with only one observation of the r.v., say 5555?
2
votes
1answer
97 views

CDF and MGF of a Sum of a discrete and continuous random variable

I am currently dealing with the following exercise: Given the random variables $X \sim Be(p), Y \sim Exp(\lambda)$, and assume they are independent. Set $Z:= X + Y$. Compute the Moment Generating ...
1
vote
1answer
101 views

The distribution of the product of a Bernoulli & an exponential random variable

Let $X$ be an exponential random variable $f(x) = c e^{-c x} \text{ if }x > 0; 0 \text{ otherwise.}$ Let $Z$ be a Bernoulli RV with $Pr(Z=1)=0.45$ and $Pr(Z=0)=0.55$. $X$ and $Z$ are independent. ...
3
votes
2answers
73 views

Prove convergence in distribution for n times the minimum of an unknown positive distribution

Let $Z_1, Z_2, ...$ be independent and identically distributed random variables with some density $f$. Suppose that $P(Z_i > 0) = 1$, and that $$ \lambda = \lim_{x\to 0} f(x) > 0$$ Let $X_n = ...
1
vote
1answer
71 views

Regression model when the dependent and independent variables show exponential distribution

As the Title suggests i am trying to figure out what would be the regression model to use when both the dependent and independent variables show an exponential distribution. Do I have to perform a ...
1
vote
0answers
68 views

Can you have an exponential distribution where x is negative? [closed]

I have a random variable with an exponential distribution and have solved an inequality to determine the maximum a posteriori rule (where if $x > \alpha$, I will choose hypothesis 1 over hypothesis ...
0
votes
1answer
164 views

How could “sum of exponential distribution is 1” be proven?

$$f(x; \lambda) = \begin{cases} \lambda e^{-\lambda x} \quad \text { for } x \geq 0 \\ 0 \quad \quad \quad \text { for }x < 0\end{cases} $$ How can I prove that the sum of probabilities under ...
3
votes
0answers
90 views

How do we show the exponential distribution has maximal entropy on R+?

I’ve been looking around the internet, and having trouble finding a demonstration that the exponential distribution is maximal entropy on R+. I’d appreciate any points in the right direction.
7
votes
1answer
1k views

Distribution of sum of exponentials

Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$. Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-...
2
votes
0answers
61 views

Proof of alternative parameterization of Weibull Survival Model

In Parametric Survival Models by German Rodrıguez (hyperlink at bottom), it is stated the Exponential and Weibull models can be parameterized in a linear form (with time parameterized as $\log(T)$). ...
1
vote
1answer
22 views

exponential distribution [closed]

The time it takes for a team to complete a certain task has an exponential distribution with mean equals to 80 hours. Given that the task was not completed in 50 hours, what is the probability that ...
3
votes
1answer
130 views

Confidence interval for shift parameter of a shifted exponential distribution [duplicate]

Define $T = \theta + X$, where $X \sim \textrm{Exp}(\lambda)$, and $\theta$ is a constant. I would like to compute a confidence interval for $\theta$ from observations $t_1, \ldots, t_n$ drawn from $...
0
votes
1answer
267 views

clarifying exponential-gamma conjugate prior

I'm referring to page 22 of this white paper. On page 22, it says the following: given that $s_i \sim \text{Exp}(\theta), i = 1,..,c$ $\theta \sim\text{Gamma}(k, \Theta)$, Then the posterior ...
0
votes
1answer
131 views

How to estimate a probability distribution for the waiting time before an event is observed

I am trying to model a type of event that happens (once) at an unknown time. I would like to know: given a certain average event time, what is the probability that the event will happen within a ...
3
votes
1answer
97 views

Probability distribution for independent time to event

I am looking at waiting times between two events from multiple patients, so I'm looking at a gamma distribution. Turns out, the model is plotting out an exponential distribution, which if I was to ...
1
vote
1answer
36 views

Extracting maximum (x,y) values and initial slope of non-linear fitted glm curves

I have plotted 8 curves using a log-link Gaussian model, y=ax exp(bx)+ϵ, for my data. I am not sure how to extract the maximum (x,y) from each of the fitted curves using the equation, of which I ...
3
votes
1answer
231 views

Double exponential decay function fitting

I have a model where I assumed an exponential increase and then decrease with a beta distribution. I fitted curves to the sets of data using the following equation: ...
0
votes
2answers
74 views

Probability of $P(X=x)$ exponential distribution

I am trying to find the waiting time which will occur with the probability of 0.99 . To do so, I do $P(X=x)=0.99$ and I did $\lambda*\exp(-\lambda*x)=0.99$ where $\lambda=0.5$. I find the $x$ value as ...
0
votes
0answers
18 views

exponential waiting time model, inflated by events that never happened?

Suppose we have some data where we see the age of each sample and whether each sample received treatment. Further, for those that received treatment some (but not all) of them have a time of treatment ...
1
vote
1answer
91 views

Is this QQ convex or concave?

With my data I got this exponential QQplot : is this QQplot convex or concave ? Thanks in advance
1
vote
0answers
388 views

Prove that the interarrival times of a Poisson Process are all indipendent and identically distributed

{$N_t$} with $t\in \mathbb{R}$ is a Poisson process with intensity $\lambda \in \mathbb{R^+}$, so that 1) $N(0)=0$, 2) {N(t) is with indipendent increments and omogeneous increments and 3) $\...
4
votes
1answer
65 views

A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables

Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
0
votes
0answers
17 views

How to model exponential decay with seasonality

I am trying to model sales driven by marketing tactics with promotional deadlines in Excel. The overall trend of the sales driven by each marketing tactic decays exponentially following the launch ...
0
votes
1answer
93 views

shapiro test for exponentiality [closed]

After running the shapiro.exp.test in testing for exponentiality I am getting powers of the test as equal to zero. I suppose the issue might be on my rejection criteria (i.e ...
6
votes
1answer
244 views

a fast uniform order statistic generator [closed]

Can someone provide me with the mathematical expression for this code/function as a fast way to generate $n$ sorted $U[0,1]$ random numbers: ...
1
vote
1answer
16 views

Different density-functions in different books for Exp(a), why?

This is more of a theoretical question, that I hope someone is willing to explain to me. I have noticed that the density function for the exponential distribution looks different in two of my books. ...
3
votes
1answer
143 views

Mean of maximum of exponential random variables (independent but not identical)

I am looking for the the mean of the maximum of N independent but not identical exponential random variables. I found the CDF and the pdf but I couldn't compute the integral to find the mean of the ...
3
votes
1answer
110 views

What is the trick in computing this expectation?

A machine starts operating at time $0$ and fails at a random time $T.$ The distribution of $T$ has density $f(t)=(1/3)e^{-t/3}$ for $t\gt 0.$ The machine will not be monitored until time $t=2.$ The ...
0
votes
1answer
480 views

Deriving Transition Matrix of the Embedded Markov Chain given the generator matrix?

Full Problem: A continuous-time Markov chain has generator matrix $$Q= \begin{pmatrix} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 2 & 2 & -4 \\ \end{pmatrix} $$ (i) ...
1
vote
0answers
39 views

Distribution of a random variable involving Exponential random variables

Let $X$ and $Y$ be two independent $\text{Expo}(1)$ random variables. Let $M:=\max(X,Y)$ and $L:=\min(X,Y)$. How do I show that $M-L\sim \text{Expo}(1)$? I have made an attempt as follows: We note ...
0
votes
1answer
29 views

how is it beneficial to Derive a formula for the maximum likelihood and apply it to a dataset

What is the benefit of applying the following on the data set? If I have a data set that is distributed as follows: after calculating the Maximum of logarithm of the likelihood function and applying ...
0
votes
1answer
68 views

Exponential Distribution Simulation Problem

I have the following problem: I am told the following information: (1) vertex A to vertex X is described by an exponential distribution with lambda = 4; (2) vertex A to vertex Y is described by an ...
1
vote
0answers
61 views

Exponential distribution as a differential equation

I'm trying to interpret the following situation. In an economy, let $T$ denote the remaining lifetime (a stochastic variable) with exponential distribution and a Cumulative distribution function ...
2
votes
1answer
46 views

Solving a marginalization integral involving exponential distributions

I'm trying to solve a marginalization integral \begin{equation} \int p(y,w) dw \end{equation} in order to compute the density $p(y)$. I assumed the following model: \begin{equation} y = (u+w)^2 + v \...
0
votes
0answers
86 views

Exponential Distribution Alarm System

I'm trying to create an alarm system where I'm able to detect when a metric is behaving abnormally. The metric is a financial one, say EPX, Earning per Unit, which means that each unit earns money ...
2
votes
1answer
5k views

Python - Test if my data follow a Poisson/Exponential distribution

my question is very close to this one and this one but I would like to have more details. I have some data and I want to check the amount of error I would have if I assume that these data follow an ...
1
vote
1answer
52 views

Determining if an event is part of a poisson process

Trying to understand how to find an outlier in a Poisson process. Using example below to help me understand. A machine creates widgets at a rate of 10 per hour ($\lambda =10$) under normal ...