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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Connection between the exponential distribution and extreme value distribution

I found here a justification that the connects the exponential distribution and standard extreme value distribution that goes as follows. Let $X\sim \text{Exp}(\lambda)$ and $Y=\log(X)$. Now $\lambda(...
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Integrating functions [closed]

Attached is an integral containing a variable (u) and products of two exponential functions. Kindly assist to proffer solution
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How to find the right scaling for exponential distribution

I have a complex Gaussian variable, $Z=X+jY$ with $X,Y \sim \mathcal{N}(0,\sigma^2)$, and I would like to find the parameter that scales the distribution of the squared magnitude $P=|Z|^2$. As ...
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Confidence Interval for exponential distribution using pivot quantity

Let' say that $X \sim exp(\theta)$. And we have a sample of size $n$ of $X$ and we consider as an estimator $\hat{\theta} = X_{(1)} = min\{X_1,...,X_n\}$ and also consider $Y = \theta X_{(1)}$. a) ...
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UMVUE for P(X > k) in exponential distribution [duplicate]

I have to find UMVUE for $exp(-k*a)$ where X ~ Exponential(a); k is a positive real number. I tried it using Lehmann-Scheffe theorem. Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
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Modeling decreasing price elasticity of a good

I am attempting to model the decreasing price elasticity/response for a good. I need to control for place and time features and available alternatives. Besides this, I also need to add time and ...
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If LASSO is equivalent to Bayesian Regression with a Laplace (double exponential) prior, what would be the prior for non-negative LASSO? Exponential?

We know that the LASSO penalty is equivalent to Laplace prior. So what would be the corresponding prior for a non-negative LASSO? Is it exponential distribution? More generally, is it true that every ...
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34 views

Find the expectation of an exponential distribution estimator

So we've got a sample data coming from exponential distribution with parameter $\lambda$, and we take an estimator $\lambda_n = \frac{n}{X_1+X_2+\cdots+X_n}$. I need to show that this is a biased and ...
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24 views

Probability of joint dependent events

I'm having trouble finding a way to do this calculation and checking if I'm correct: Let $X_1 \sim Exp(2)$ and $X_2 \sim Exp(2)$ be independent random variables $\left(f_X(x) = 2e^{-2x}\right)$, ...
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Expected Waiting Time for Each Person in a Queue

Suppose I have the following problem: There is a coffee shop with 2 employees (server) taking orders. Customers come to the coffee shop and form a queue, based on "first in, first out". ...
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Insurance Claims: Proving a Process is a Poisson Process and Finding its Rate

Let $X(t)$ denote the number of claims received by an insurance company in the time interval $[0,t]$. We will assume that ${X(t) : t ≥ 0}$ can be modelled as a Poisson process, where $t$ is measured ...
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Interpolation: More advanced calculation than linear regression

I am currently modelling the customer growth of a company. I have assumed values for December 2020, 2021, 2022 and 2023 and want to estimate the values for the month in between. As you can see in the ...
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Which is the covariance matrix between X ~ exponential (lambda) and X squared?

I need to find the covariance matrix between $X \sim exp(\lambda)$ and its square $X^2$. Can I interpret $X^n$ as an Erlang distribution (for which I can have the expectation/variance on Wikipedia :-) ...
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104 views

The random variable $log(\frac{X}{x_0})$ has an exponential distribution with parameter $\alpha$

It is said that a random variable $X$ has a Pareto distribution with parameters $x_0$ and $\alpha$ for $(x_0 > 0)$ and $(\alpha > 0)$ if $X$ has a continuous distribution for which the p.d.f. $f(...
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Bernoulli process and two exponentials

Suppose that a very long Bernoulli process gives a sequence with possible values: $A$ with probability $p$, and $B$ with probability $1-p$. The expected fraction of contiguous sequences of length $k$ ...
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CLT application to exponential distribution? [duplicate]

I'm a little confused how the CLT can apply to aggregations of the exponential distribution. It's my understanding that the CLT says, in plain English, "sample means from virtually any ...
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Calculating Average Days Between Customer Orders - Exponential Distribution?

I have detailed sales data including customer number, SKU, product line, and order date. For marketing purposes, I would like to know the average days between orders both on a per-customer, per ...
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1answer
198 views

What is area under cumulative distribution represent? [duplicate]

Exponential distribution has following probability density function which explains the curvature of a line (For simplicity I am just going to work with x>=0): f(x) = lambda e^{-lambda*x} to find ...
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When can taking the following transformation of data $log_{10}(e^{mean(log(x))})$ be useful?

I wished to transform some data and have noticed that the geometric mean is a known transformation to normalise data with high differences in comparably large values relative to lower values. However, ...
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Consistency of $f( x| \theta ) = \exp(-(x- \theta ))$

Prove that the second smallest observation in a random sample of size n from following pdf is consistent estimator of $ \theta $ $$ f( x| \theta ) = \exp(-(x- \theta )) , \qquad x > \theta $$ ...
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91 views

Expected Value of Exponential CDF

I am given the following CDF and I want to calculate its expected value: $F(Y \leq y) =1-( 0.28e^{-0.5y} + 0.71e^{-0.25y})$ Creating the PDF: $f(Y \leq y) = \frac{71\mathrm{e}^{-\frac{x}{4}}+56\mathrm{...
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Generating a Random Value Vector from an Exponential Distribution using R

Given a standard PDF of the form $f(x)=ae^{-ax}$ with domain $[0,+\infty)$, its CDF being $F(x)=1-e^{-ax}$, and a mutated CDF that takes $p \in [0,1]$ as a probability and returns the corresponding $x$...
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Exponential decay

I have a new product that was launched and I see a bug spike in enrollments during the initial stages post which it's been declining and now it holding steady. Curve looks like an exponential ...
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How do I test for difference in means of two exponentially distributed samples, when all I have is mean, standard deviation, and sample size? [duplicate]

I have two samples, and in each sample I measured fluorescence intensity from 10k cells. Unfortunately the instrument doesn't return raw data, just histograms and summary statistics (mean, st.d., and ...
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96 views

Expectation Values of a Binomial Distribution with an Exponentially Distributed Variable

I have an exponentially distributed variable, $Y$: $Y \sim Exp(\lambda_{0}, x) $ I want to calculate the expectation value of a binomial distributed variable $Z$ dependent upon the value of $Y$ given ...
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Bartholomew estimate of variance of median for exponential distributed RVs

Suppose, $X_1, X_2, \ldots X_n \sim \text{iid Exponential}(\theta)$. The median is given by $\log(2) \theta$. The MLE of the median is given by: $$ \hat{M} = \log(2) \sum_{i=1}^n X_i / n $$ And the ...
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132 views

Prove that the sum of exponential random variables is a gamma distribution [closed]

I tried to prove using the convulution approach but it didn't work
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Finding the likelihood function of parameter vector

I am trying to do an excerise on likelihood functions and score vectors and I am having some trouble extending it to parameter vectors. The question is stated below. Suppose we have two independent ...
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Simulating a non-simple exponential distribution

I want to simulate a vector x in R with 1000 entries. The j'th entry comes from a exponential distribution with density $$ f_{j}(x)=j \beta e^{-j \beta x}, x>0 $$...
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exponential parameter estimtion from the smallest k-th order statistics

Assume $X_1, X_2, X_3,\ldots,X_n$ are i.i.d. samples from Exp($\lambda$). Assume that the integer $k<n$, is it possible to find a an unbiased estimator for $\lambda$ from the k-th smallest ordered ...
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Finding UMVUE of difference of exponentals

Let $X_1, \ldots, X_n$ be a sample from an exponential distribution with p.d.f. $f(x; \theta) = \theta e^{-\theta x}$ for $x > 0$ where $\theta > 0$ is an unknown parameter. I would like to find ...
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AUC as Summary Statistic for Large Data Sets

Let's say I have a couple of treatment groups with 100 subjects each. Each subject generates thousands of data points that do not conform to a "nice" overall distribution. But if I sort the ...
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How do I generate random integers within a specific range in R?

I want to generate 100 random integers from an exponential distribution in R, where each integer is between 10 and 50 (hence 10, 11, 12, ..., 49, 50). How can I do this in R?
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Using exponential distribution to generate loads for user for load tests

If i understood the exponential distribution correctly, it says the longer we wait the higher/lower probablity of event to occur there is. So for example if we wait for a bus, the longer we wait the ...
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Likelihood Ratio for two-sample Exponential distribution with constant ratio and confidence interval

Let $X$ and $Y$ be two independent random variables with respective pdfs: $$f \left(x;\theta_i \right) = \begin{cases} \frac{1}{\theta_i} e^{-x/ {\theta_i}} \quad 0<x<\infty, 0<\theta_i< \...
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The exponential distribution belongs to the exponential family [closed]

I'm new here. I'm trying to proof that the exponential distribution belongs to the exponential family, but I don't know how to do that. Can you help me? Thanks a lot.
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The expected value of log Gamma function

Suppose $X$ is exponentially distributed with the rate parameter $\lambda$. If we have the expected value of $\log X$ as \begin{equation} \langle \log X\rangle=-\gamma-\log\lambda \end{equation} where ...
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Memoryless property: how is $F^c(2) = F^c(1)F^c(1) = (F^c(1))^2$ and $F^c(1/2) = (F^c(1))^{1/2}$ implied?

I am currently studying the textbook Modeling and analysis of stochastic systems, third edition, by Kulkarni. Chapter 5.1.1 Memoryless Property says the following: We begin with the definition of the ...
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59 views

How to correctly transform a log-log graph into untransformed exponential graph?

I plotted my data on a natural log-log scale and I seem to get a okay fit to the data with y=1.19 - 0.116x with Rsq = 0.29 I want to use the parameters but plot the row data with an exponential curve....
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Memoryless property: how does $P(X > s + t \mid X > s) = P(X > t), s, t \ge 0$ imply that $F^c(s + t) = F^c(s) F^c(t), s, t \ge 0$? [closed]

I am currently studying the textbook Modeling and analysis of stochastic systems, third edition, by Kulkarni. Chapter 5.1.1 Memoryless Property says the following: We begin with the definition of the ...
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1answer
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How r sample from exponential distribution?

I managed to find the source code in sexp.c, and the algorithm (Ahrens & Dieter). I mostly understand the first half of the code - it seems like it finds the ...
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289 views

Coefficient of variation for exponential distribution: $\text{Var}(X)/E(X)^2$?

I am currently studying the textbook Modeling and Analysis of Stochastic Systems, third edition, by Kulkarni. Chapter 5.1 Exponential Distributions says the following: The probability density ...
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Optimising exGaussian loss

I have been using the PyTorch class torch.nn.GaussianNLLLoss for an optimisation problem that I have. It optimises the mean ($target$) and variance ($var$) of a ...
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Most Powerful test for indicating exponential or Weibull distribution [closed]

Let $X_1,...,X_n$ be iid distribution function of $F(x)$. I want to test whether $F$ is exponential or Weibull. This means that either $F(x)=1-exp(-x), x>0$ (exponential) $F(x)=1-exp(-x^{\theta}), ...
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Why do we associate the negative symbol in the exponent of $e^{- \theta y}$ to $T(y) = -y$ rather than $c(\theta) = \theta$?

I am currently trying to learn about the exponential family of distributions. I have the example $Y \sim \exp(\lambda)$, where the density is $f_\theta (y) = \theta e^{- \theta y}$ for $y \ge 0$ and $\...
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Enforcing conditions on truncated exponential distribution

The CDF for an exponential distribution of rate $\lambda$ truncated at T is $F(t) = \frac{1-e^{-\lambda t}}{1-e^{-\lambda T}}$. (for $t<T$, else 0). I would like to determine $\lambda$ and $T$ such ...
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How is this implied by the properties of the exponential, gamma, and $\chi^2$ distributions?

Let's say we have the random variables $X_1, \dots, X_p$. Furthermore, say that these random variables are a random sample from a PDF of the form $$f_\tau (x) = \begin{cases} \tau x^{\tau-1}, & 0 ...
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Complete sufficient statistic of non-identical distribution: $X_i \sim EXP(i\theta)$

Problem Suppose that $X_1, \dots, X_n$ are independent $\mathrm{EXP}(i\theta)$ random variables. Find a complete sufficient statistic for $\theta$. My Attempt Since pdf of $x_i$ is \begin{equation} ...
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Expected value of (continous) exponential distribution proof/derivation

I started with the following exponential distribution: $$ f_{exp}(x;\lambda) = \lambda\, e^{-x\lambda} \quad \forall\, x \in \mathbb{R}^+ $$ I know from internal courseslides and wikipedia that the ...
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76 views

Memorylessness of exponential and expectation

Suppose I have a teller who has servicing time that is exponential with mean of $2$ minutes. Say customer $A$ arrives at noon and begins being serviced by the teller. What is the expected length of ...

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