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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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Exercise about Order statistics from uniform distribution

I'm trying to solve an exercise about order statistics. The exercise is the following: Let $U_{(1)}< \ldots <U_{(n)}$ be the order statistics from Uniform distribution U(0,1). Show that $(-\log[...
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What is the distribution of needed hospital beds?

Suppose I am modelling a hospital service with $k$ number of beds. Initially there are $m$ number of beds being used, where $m \leq k$, each of which has a known amount of time that it has been ...
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How Do I Calculate the Scaled Deviance of a GLM with Gamma(Exponential) Distributed Dependent Variable?

I'm fitting a generalized linear model to a theoretically exponentially distributed dataset. The exponential distribution has PDF $$ f(y;\lambda) = \lambda e^{-\lambda y} $$ This question Deviance for ...
Jack Guan's user avatar
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Conditional Likelihood exponential distribution [closed]

Let $X_1, ... X_n$ be iid Exp($\lambda$), where $\lambda > 0$. How does the Maximum Likelihood Estimator (MLE) of $\lambda$ change if we somehow are told that all $X_i$ overshot their mean? (i.e.: ...
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The Probability of an Earthquake Event

I'm looking for ways to to find the probability of an earthquake event from Twitter posts. I came across an equation in a research article that I need to understand and use. My goal is to write a ...
ORDeSUrv's user avatar
1 vote
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The distribution of K until the n-th occurance of a poisson process takes place

Let's take the following scenario: Customers arrive at a shopping mall according to a Poisson process with a rate of λ per minute. The mall has a door that closes ...
Sandra Sukarieh's user avatar
1 vote
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Link function for exponential regression

I have a dataset with an exponential relationship that I'm linearizing before fitting with code like this: glm(log(y) ~ log(x), family = gaussian) I'm trying to ...
tnt's user avatar
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Modeling data with continuous and discrete measurements

I am working on a project involving a dataset which includes variable which represents "time spent", but the data is a combination of two measurement sources: one is discrete and one is ...
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Distribution of videos watched, given people drop out from course

An online training course involves 100 videos, which people watch one per session, starting with 1 and finishing at 100. People drop out of the course, having completed N sessions, and watched videos ...
Derek Jones's user avatar
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1 answer
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Posterior probability for $\theta$ with a discrete prior

I'm trying to find a posterior probability for this model but I can't find the solution. Help would be appreciated! Prior distribution: $\theta$ follows a discrete probability function: $\mathbb{P}(\...
Alexandre Beaudry's user avatar
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1 answer
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Confidence interval on ratio of estimates for exponential random variables

Given exponential random variable X, the MLE for the scale parameter is $\hat{\beta_x} = \bar{x}$, and the confidence interval for that estimate is: $$\frac{2n\bar{x}}{\chi^2_{\frac{\alpha}{2},2n}} &...
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Measure Theoretic Justification For Manipulating Conditional Probability of Events Given a Continuous Random Variable and an Event [closed]

When considering the conditional probability of an event given a continuous random variable (or vector) can you essentially just manipulate the probability as if you were only working with discrete ...
PerpetuallyConfused's user avatar
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Maximum Entropy distribution of a ticking clock

Say I have a clock that emits "ticks". An ideal clock looks like a dirac comb. It has: perfect periodicity of ticks (there is a precise fixed time interval between any two consecutive ticks)...
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How would I determine statistical significance on exponential data?

I'm trying to determine if there's a statistically significant difference between the O2 permeability into two different materials. The measurements are percentage of O2 (O2%), measured every 15 ...
Geordan Nicholson's user avatar
1 vote
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Showing that $X_{(1)}$ is sufficient for shifted exponential distribution

If the pdf of a random sample is $f(x)=e^{-(x-θ)}$ where $x \geq θ$, Show that $T=X_{(1)}$ is a sufficient statistic for $θ$. Can one show that $T$ is a sufficient statistic for $θ$ in the following ...
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`pgeom` in `R` does not seem to exhibit memorylessness as expected

The geometric and exponential distributions exhibit memorylessness, i.e., $$ P(X \geq k + j)=P(X\geq k)P(X\geq j) $$ This can be shown for an arbitrary exponential distribution using ...
mazin-abdelghany's user avatar
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Logit probability with many option closely approximated by exponential probability

In the paper "Firms and Labor Market Inequality: Evidence and Some Theory" by David Card and coauthors (2018) I find this passage, where a logit choice probability is closely approximated by ...
Francesco Armillei's user avatar
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What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?

I had asked this question on Math stackexchange once before and now again but this does not seem to be drawing too much attention. Since this is a question that can be safely classified as non-measure ...
Dovahkiin's user avatar
4 votes
2 answers
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The density of the sum of independent product of Bernoulli and Exponential

Let $X_1,\cdots,X_n$ and $Y_1,\cdots,Y_n$ are iid from $X\sim \operatorname{Exponential}(\lambda)$ and $𝑌\sim\operatorname{Bernoulli}(p)$ respectively, where $X$ and $Y$ are independent. Given $n,\...
John Stone's user avatar
5 votes
1 answer
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Inferring distribution over concurrent events given arrival and duration distributions

Suppose you're working on infrastructure for web service and you're tasked with determining the probability distribution of concurrent requests to your services. In example, suppose the first request ...
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Proving that the average number of arrival events as $\lambda t$ given the inter-arrival duration are i.i.d. Exp($\lambda$) random variables

I'm trying to prove a common result for the Poisson process but I'm stuck. Given $T_i$ are i.i.d. $Exp(\lambda)$ random variables (where $\lambda$ is the rate) that represent the duration of arrival ...
Khai Yi Chin's user avatar
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How to find the MGF of the max of a set of i.i.d. exponential random variables

As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
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How to apply the antithetic variates method for an exponential distribution?

I'm trying to complete an exercise in R related to Monte Carlo estimation for the shortest path in the bridge network problem. The exercise asks me to first perform the estimation using the plain ...
Mr Economics's user avatar
11 votes
1 answer
221 views

Distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$ when $X_i$'s are i.i.d $\text{Exp}(1)$

Suppose $(X_n)_{n\ge 1}$ is a sequence of independent Exponential random variables with mean $1$. I am trying to find the distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$. Simulation suggests the ...
StubbornAtom's user avatar
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Estimating survival curve with changepoint from Poisson regression

Suppose I fit data according to the following piecewise exponential model. So that the intensity is 1/50 for event times in the first 50 days and then 1/200 subsequently. ...
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3 votes
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Cullen and Frey graph can't recognize exponential distribution

I am trying to familiarize myself with the Cullen and Frey plot. So, I generated 1000 exponentially distributed numbers using the command: x <- rexp(1000, 2) ...
Kώστας Κούδας's user avatar
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1 answer
183 views

In R Limma's `normalizeMedianValues` function, why is its operations preceded with a log transformation followed by an exponential transformation?

In the function normalizeMedianValues in the package limma, column counts are normalised such that their column medians are ...
SpikyClip's user avatar
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Relationship between Poisson count and Exponential inter-arrival times [duplicate]

This picture represents transactions. $T_1$ is the time of the first transaction, $T_2$ time of second, $T_x$ time of the $x$th transaction $t$ is the observation period, $X(t)$ denotes the number of ...
statmath30's user avatar
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Solve equation for a given set of parameters

I'm modelling a process for which the probability of the event (stop) not happening before time $t$ is $e^{-\lambda\cdot t }|\lambda>0$. When the event happens, the process stops running for a ...
Jon Nagra's user avatar
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Let $N(t)$ be a Poisson process, compute $P\{N(s)=1,N(t)=2\}$ for any $0\leq s<t$

I got the answer as $\lambda^2e^{-\lambda t}s(t-s)$ using the properties like independent increment and stationary increments. But I can't seem to understand the steps in the solution of the book. For ...
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Hypothesis Testing via Chi Squared

I am having trouble understanding the application of chi square "power of the test" in context to hypothesis testing. The following is not homework, I'm simply providing an example to convey ...
Harry Ferrier's user avatar
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Question about large estimated standard error on exponential distribution

I have collected data that is the time between discrete random events and I have a parameter $V$ (voltage) that changes the mean rate of my events. I have used MLE to fit the exponential distribution ...
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1 vote
1 answer
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Probability distribution of actual time spent if randomly sampled at a known mean rate

I was experimenting with tagtime, which randomly asks the user what they're doing at a known mean rate $\lambda$. Let's say that every time I am sampled, I give a yes/no answer. If I answer yes $k$ ...
ttttcrngyblflpp's user avatar
1 vote
1 answer
233 views

Correctly simulating an extreme value distribution for survival analysis?

In the image and per the code at the bottom of this post, I plot survival curves for the lung dataset from the survival package using a fitted exponential ...
Village.Idyot's user avatar
2 votes
1 answer
106 views

How to appropriately model the uncertainty of the exponential distribution model when running survival simulations?

In the code shown at the bottom of this post, I plot survival curves for the lung dataset from the survival package using a ...
Village.Idyot's user avatar
1 vote
1 answer
157 views

Confirm that the exponential distribution is correctly being used with the survreg() function of the survival package?

I've been examining fitting the Weibull and lognormal distributions with the survreg() function of the survival package. Fitting ...
Village.Idyot's user avatar
1 vote
0 answers
18 views

Fitting a set of points to a distribution by adding up to three degrees of freedom with Python [closed]

I have a set of points whose shape is as below: Its set of x and y points is as follows: x=[0.14741,0.180288,0.195,0.245342,0.25614,0.289377,0.315789,0.357143,0.431034,1.785714,2,2.323529,2.586207,3,...
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Are these data points decaying exponentially or as a power law?

0 I have a set of data points. The first coordinate is time and the second coordinate is energy. I am trying to figure out how the energy is decaying over time. Particularly, I have to find if it is ...
HadamardN2's user avatar
2 votes
1 answer
138 views

Testing relationship between exponential and beta distributions using R

If X ~ Exp(3), Y ~ Exp(1) and h = X / (X + Y) then h ~ beta(1/3, 1) and E(h) = 1/4. But when I draw random deviates using the following R code, I find mean(h) ≈ 0.324 and the histogram doesn't ...
rob's user avatar
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0 answers
25 views

What is the approximate distribution of the distance between gas stations along a highway?

Thinking it should be exponential, but I could be wrong.
Mazen Danaf's user avatar
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1 answer
60 views

Why the actual hazard ratio of the simulated time-to-event data is very different from the expected value?

I am trying to generate time-to-event data for two treatments, with 25 patients in each arm. I generated the survival time in the treatment arm using an exponential distribution with parameter 0.95, ...
Kana's user avatar
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5 votes
2 answers
238 views

Hypoexponential distribution is stuck because of $\lambda_i \neq \lambda_j$?

The definition of the hypoexponential distribution (HD) requires that: $$f(x)=\sum_i^d \left(\prod_{j=1,i\neq j}^{d}\frac{\lambda_j}{\lambda_j-\lambda_i}\right)\lambda_i e^{(-\lambda_ix)},\quad x>0 ...
Oliver Amundsen's user avatar
1 vote
1 answer
113 views

Determining statistical significance on exponential variables and timeseries

I am attempting to determine statistical significance of various customer programs for farmers use of sustainable products. Farm sales of sustainable products follows roughly an exponential curve, ...
WolVes's user avatar
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2 votes
1 answer
159 views

Poisson Process: Probability distribution to describe time (distance) to successful event?

Given some length of time $t$ with successful events occurring in this interval at rate $\lambda$. Assume that only one successful event occurs during this interval of length $t$. Which distribution ...
Nic's user avatar
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5 votes
1 answer
246 views

The natural parameterization of the exponential class of densities always exist?

Def. Exponential Class of Densities The density function $f\left ( \mathbf{x};\mathbf{\Theta} \right )$ is a member of the exponential class of density functions iff $$f\left ( \mathbf{x};\mathbf{\...
Elisa's user avatar
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1 vote
1 answer
300 views

Probability that one of two exponential random variables is the smaller

Let $X_1 \sim \text{exp} \left( {\lambda}_1 \right)$ & $X_2 \sim \text{exp} \left( {\lambda}_2 \right)$, and they are independent. Now consider the random variable $Y = \min \left[X_1, X_2 \right]$...
Brian Smith's user avatar
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59 views

Convergence in probability of $n \cdot \min _{1 \leq j \leq n} X_j$?

Suppose $X_1, X_2, \ldots \sim U(0,1)$ and $X \sim \operatorname{Exp}(1)$, and $X_1,X_2, \dots , X$ are independent. Does it follow that $n \cdot \min _{1 \leq j \leq n} X_j \stackrel{P}{\rightarrow} ...
Eric Blyth's user avatar
2 votes
0 answers
72 views

Obtaining MLE of the parameter of exponential distribution

Let say I have a sample of size $n$ as $\{X_1,X_2,...,X_n\}$. The sample points $X_i$ are integers, but each of them are actually integer ceiling of corresponding real number $\{Y_i\}$. For example if ...
Bogaso's user avatar
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2 votes
1 answer
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Find nth central moment for exponential distribution

I am trying to figure out what the nth central moment is for the exponential distribution. Here is the formula for the nth moment: $$ \mathop{\mathbb{E}}{[x^n]} =\dfrac{n!}{\lambda^n} $$ My question: ...
Alexander Mills's user avatar
4 votes
1 answer
158 views

Calculate the distribution of the sum of mass arriving in random intervals over some period

Say we have a pipe spitting out potatoes. The potato masses are distributed with exponential distribution $$ P(\textrm{potato mass}= x)=\lambda_1e^{-\lambda_1x}$$ Each potato arrives with intervals ...
extracrispy's user avatar

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