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Questions tagged [exponential-distribution]

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

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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 ...
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42 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 ...
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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 ...
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Bayesian approach for exponential and Cox proportional hazard model in R

I have difficulty understanding and applying the Bayesian approach to survival analysis. I assume that data follow exponential life time likelihood and that failure rate lambda can be expressed in ...
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50 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?
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59 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 ...
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64 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. ...
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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 = ...
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45 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 ...
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52 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 ...
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99 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 ...
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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.
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449 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}^{-...
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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 $\log(T)$). ...
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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 ...
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1answer
84 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 $...
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117 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 ...
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1answer
121 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 ...
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1answer
63 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 ...
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1answer
34 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 ...
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1answer
179 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: ...
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60 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 ...
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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 ...
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1answer
62 views

Is this QQ convex or concave?

With my data I got this exponential QQplot : is this QQplot convex or concave ? Thanks in advance
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197 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) $\...
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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 $...
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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 ...
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1answer
75 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 ...
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1answer
231 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: ...
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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. ...
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1answer
104 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 ...
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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 ...
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342 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) ...
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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 ...
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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 ...
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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 ...
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Proving Consistency of 'theta' for an exponential distribution

$X_1, X_2, \dots X_n$ constitute a random sample of size $n$ from an exponential distribution: a) show that $\bar{X}$ is a consistent estimator of the parameter $\theta$ b) with reference to part a,...
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56 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 ...
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1answer
45 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 \...
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23 views

Estimating change in probabilities in 2 exponential distributions

I have 2 exponential distributions with different parameters. The distributions represents probabilities of 2 classifiers. Given the known distributions and a probability of an item from 1 classifier, ...
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0answers
70 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 ...
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1answer
4k 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 ...
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1answer
47 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 ...
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1answer
72 views

Variance of log($X_{\lambda}$) where $X_{\lambda}$ is a exponential function? [duplicate]

Let $X_{\lambda}$ has exponential density $\lambda e^{-\lambda x}$. Then what is the nature of Var log($X_{\lambda}$) w.r.t $\lambda$ i.e. is it increasing, decreasin, etc on $\lambda$. My approach: ...
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Lambda - Exponential vs. Poisson Interpretation

I'm trying to understand $\lambda$'s role in both the Poisson and Exponential Distributions and how it is used to find probabilities (yes, I have read the other post regarding this topic, didn't quite ...
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Regression of a linear combination of exponential decay functions

I have $n$ data points that I would like to model via somewhat of a classical linear regression model like this, where $i$ is one of the $n$ data points: $w_1x_1^{i}+w_2x_2^{i}+...+w_mx_m^{i} = y^{i}$...
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126 views

Interpreting coefficients in AFT model

I'm using an AFT (accelerated failure time) model with an exponential distribution to study the speed of adopting a new innovation. One of my independent variables is cumulative adoption, which is ...
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32 views

How random variable $Y$ following geometric distribution in the following situation

I have a random variable, $X$ which follows exponential distribution with parameter $\lambda$. Then, define $Y=k$ for some $a$ greater than zero such that: $ka \leq X \leq (k+1)a$. Now, my ...
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1answer
70 views

Exponential distribution — calculating probability that computer works!

If X is the number of years a computer works, and it follows an exponential distribution with a lambda 3, what's the probability a computer will work in 8 years? I'm not sure that I'm going about ...
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1answer
252 views

How to generate random samples from Gumbel’s bivariate exponential distribution?

The earliest and the simplest known bivariate exponential distribution, introduced by Gumbel (1960), has joint survivor function and joint probability density function given by: \begin{equation}\...