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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Probability that the expected value (mean) is within one standard deviation of an exponential distribution [closed]

fx = 1/5e^-x/5 x>0 Calculate the probability if the mean is within one standard deviation. What I Know: mean = 5 (because it is theta or the inverse of lambda from the fx) In an exponential ...
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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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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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Likelihood Function for exponential Distribution [duplicate]

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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56 views

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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127 views

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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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
134 views

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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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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1answer
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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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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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1answer
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What type of model(s) can I use for my (definitely not linear) dependent variable?

I have dependent variable, measured with a range of 0-100% (nevertheless it takes on fairly few variables). It reflects the amount of sales reported for some purpose. The distribution looks as in the ...
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How does one define the sum of N random variables in Python? [closed]

Given $X_1 \cdots X_n \stackrel{iid}{\sim} exp(1)$ I want to show that $Y = 2\sum_{i=1}^{n}X_i \stackrel{}{\sim} \chi^2_{2n}$ I proved it by computing the MGF of Y as $M_{Y_1}(t) = M_{2\sum_{i=1}^{n}...
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Finding bound for type 1 and type 2 error [duplicate]

Suppose that $T_1,…,T_{10}$ are iid $Exp(λ)$ and the goal is to test if $H_0:λ≤1$ versus $H_a:λ≥2$. Suppose that the test statistic is $S=\sum_{i=1}^{10}Ti$, and rejection of the null occurs when $S≤7$...
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Minimum sample size required question

The lifetime (in years) of a device, $X$ can be modelled by an $Exp(1)$ distribution. What is the minimum number of devices I need so that, with probability no smaller than 0.95, at least 30% of ...
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Is there a name for this generalisation of the exponential distribution

Is there a name for the following: $$ f(x) = \lambda(x) e^{\int_0^x -\lambda(t) dt} $$ which is similar to an exponential distribution. If $f(x)$ is a polynomial, would this be classed as a gamma ...
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Waiting time for specific event in multiple independent poisson processes

If I have $n$ independent Poisson processes with rates $r_1, r_2, \ldots, r_n$, and $\sum_{i=1}^n r_i = r_{tot}$, the expected time until any event occurs is $\frac{1}{r_{tot}}$. If I am interested in ...
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Maximum a posteriori estimate with exponential prior

Lets say that I have N observations that are poisson and i.i.d. The prior is an exponential with parameter 2. I know that the exponential distribution is given by $ \lambda e^{(-\lambda x)} $ But how ...
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Should you multiply every observation with the prior when calculating the maximum a posterior?

Lets say I have a number of observations and a prior. The observations are poisson distributed, are i.i.d and the prior is exponential with paramater 2. I want to calculate the maximum a posterior ...
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From two-state Markov Chains to Mean Time Between Failures / To Repair

Let $\mathcal T = \{1,2,\ldots,T\}$ denote the set of points in time, $S = \{0,1\}$ the state space, $X = (X_t)_{t \in \mathcal T} \in S^\mathcal T$ a time series, $\alpha = \mathbb P(X_{t+1} = 0 \mid ...
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For data that follows an exponential trend, is there a single equation to relate standard deviation, mean, and sample size?

Essentially, is there an equation that consists of mean, sample size, and standard deviation (or standard error of the mean) whereby I could know two of the three values and calculate the third ...
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calculating b from the Gutenberg-Richter distribution [closed]

I'm trying to calculate b from the Gutenberg-Richter distribution though, I'm struggling in understanding the calculation. For example, the equation is this: $Log_{10}N(M) = a-b(M-M_1); M \ge M_1$ $...
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Proving that $\frac{1}{\Gamma(r)}\int_{\mu}^{\infty}t^{r-1}e^{-t}dt=\sum_{x=0}^{r-1}\frac{e^{-\mu}\mu^x}{x!}$ [duplicate]

$$\frac{1}{\Gamma(r)}\int_{\mu}^{\infty}t^{r-1}e^{-t}dt=\sum_{x=0}^{r-1}\frac{e^{-\mu}\mu^x}{x!}$$ What I have tried- $$\frac{1}{\Gamma(r)}\int_{\mu}^{\infty}t^{r-1}e^{-t}dt=e^{-\mu}\sum_{x=0}^{r-1}\...
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Integral of cdf times pdf is a probability?

Let $X$ be a random variable with distribution function $F_X$. Consider $$P=\int_0^\infty (1-F_X(x))e^{-x}dx.$$ Because $1-F_X(x)$ is the probability of $X>x$ and $e^{-x}$ is the pdf of an ...
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Fisher Information invariant by a specific reparameterization of the Exponential Distribution

The exponential distribution can be parameterized in two common ways: $$ f(x) = \lambda \exp(-\lambda x) $$ where $E[X] = \frac{1}{\lambda}$ $\text{Var}[X] = \frac{1}{\lambda^2}$, or as $$ f(x) = \...
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Appropriate exponential goodness of fit test for unknown parameter

I have a set of univariate data (n=1000) that I believe to be exponentially distributed. I also have a separate, but inexact, prior estimation for the mean or parameter of the distribution I expect ...
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1answer
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Poisson variate corresponding to the Exponential variate

According to Wikipedia, In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in ...
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What relation have the Markov Property with Queueing Theory?

What relation have the Markov Property of Exponential Distribution, with Queueing Theory? i need to know the utility relation between Markov Property and Queueing Theory edit: its open to a better and ...
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483 views

Generating random samples obeying the exponential distribution with a given min and max

Random samples obeying the exponential distribution can be generated by the inverse sampling technique by using the quantile function of the exponential distribution: $$ x = F^{-1}(u) = - \frac{1}{\...
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UMVUE of two-parameter exponential family distribution

Suppose $\{X_{i}\}_{i=1}^n\overset{i.i.d}{\sim}X$, where $X$ has density $$f_{X}(x)=\frac{1}{b}\exp\left\{\frac{x-a}{b}\right\},x>a$$ What is the UMVUE of $\mathbb{P}(X_1<u)$? Here is what I've ...
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Prove $Y_2 - (Y_1 - 1)^2/2$ is an $Exp(1)$ random variable

I am studying Monte Carlo simulation and I came across with this claim from notes: Let $Y_1$ and $Y_2$ be two independent $Exp(1)$ random variables. We accept $Y_1$ as one sample when $Y_2$ is larger ...
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1answer
149 views

Minimal sufficient statistics for 2-parameter exponential distribution

Suppose $X_1, \ldots, X_n$ is a random sample with pdf $$f_{X_i}(x_i \mid \alpha, \beta) = \beta^{-1} \exp \left(\frac{-(x_i-\alpha)}{\beta}\right) I(x_i \geq \alpha)$$ for all $i = 1, 2, \ldots, n; \ ...
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1answer
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$X\sim\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. Find UMVUE of $\frac{a}{b}$

Suppose $\{X_i\}_{i=1}^n\overset{i.i.d}{\sim}X,$ and $X$ has density $f(x)=\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. What is the UMVUE of $\frac{a}{b}$? Here is what I've done so far. It can be ...
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1answer
43 views

Identify Poisson or Exponential Distribution and determine lambda

I am trying to identify the distribution of my variable, $X$. It measures goals per minute of soccer players. Possible values are $[0,inf]$ and they are non integers. I believe this to be an ...

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