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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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Show that $nX_{(1)}$ is not consistent

Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
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What are some intuitively pleasing examples of “story”-inspired derivations of a Weibull distribution? [on hold]

What are some intuitively pleasing examples of "story"-inspired derivations of a Weibull distribution?
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Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
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Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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How to compare two statistical distributions with unavailability measurements?

I have two different measuring instruments to evaulate if an electronic device is working or not. These instruments provide a working/not-working reading each day and at the end of the month I compute ...
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Hypothesis test for composite null hypothesis of exponential parameter

I'm having trouble defining the reject region based on the generalized likelihood ratio test. This is from a question of past exam I'm self-studying and still have the doubt, given that I got it wrong....
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Re-scaling of exponentially distributed random numbers

I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$ (for simplicity, $N=1$). I found that the generated numbers are initially exponentially ...
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How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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Why knowing information about something doesn't help when it comes to Exponential Distribution?

I've recently learned about Exponential Distribution and when it's appropriate to be used. There was the following example given. The duration of IC555 integrated circuits under normal operation ...
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Mann-Whitney-U for exponential distributions?

I have two distributions, that look like the following: They appear to be exponential and have different sample sizes (369 vs 60). I would like to do some hypothesis testing. I know that I can use ...
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Questions regarding this derivation of the Poisson Distribution from exponential densities

On page 217 - 218 of the pdf of this book, the author derives the Poisson Distribution using gamma and exponential densities. The author defines $S_n$ to be the sum of a sequence of independent ...
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If there are covariant variables in the exponential distribution, how do I draw the QQ plot?

Observation Data X=x[1],... x[n]; covariant variables are eta=eta[1],...,eta[n]. The parameters of exponential distribution are composed of covariant variables(eta) and lambda. So, X[i]<-rexp(eta[i]...
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To what distribution it's similar? Looks like an exponential but it's not

To what distribution it's similar? Looks like an exponential but it's not. It's seems to have a property, that if I zoom it (xlim) then each time it has the same ...
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Can Kernel Density Estimation estimate an Exponential Distribution?

Can Kernel Density Estimation estimate an Exponential Distribution? I tried to performed to make experiments with various kernels like: "gaussian" and "exponential", but performance seems to be very ...
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How do we build a confidence interval for the parameter of the exponential distribution?

EDIT Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\text{Exp}(\theta)$, where $\theta$ is not known. Precisely, $f(x|\theta) = (1/\theta)\exp(-x/\theta)$ Describe ...
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Why can't I write $P(X>5|X>1) = P(X>5)$? [duplicate]

I have a confusion with the memorylessness property of exponential distribution. If exponential distribution is memoryless (i.e. the past has no bearing on its future behavior), why can't I write $P(...
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exponential RV til bus arrives

Suppose that you are waiting at a bus stop. The waiting time until a bus arrives is $T$ where $T$ is an exponentially distributed random variable with parameter $λ$ i.e. $P(T≤t)=1−e^{−λt}, ∀t≥0$. (a) ...
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Geometric distribution described with rate parameter

I don't understand this sentence from this paper (around equation $5$): The function $H(\tau)$ is the hazard function. $H(\tau) = \frac{P_{\text{gap}}(g = \tau)}{\sum_{t=\tau}^{\infty} P_{\...
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What's the relationship between Pareto shape parameter (alpha) and exponential rate parameter (lambda)?

I'm trying to do my undergraduate research on non parametric density estimation for a heavy tailed distribution. For that, I'm with a data set, which I assumed it should be Pareto distributed with ...
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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 ...
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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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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 ...
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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
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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 $t = 7$ ...
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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 \...
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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. ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...