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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26 views

Is it possible to do Double Log Transform?

I have a data with only one parameter (i.e. y~x) and my goal is to find the correlation between the two values. Currently, the data looks something like this: I believe this data follows an ...
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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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10 views

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
16 views

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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1answer
13 views

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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387 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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52 views

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
46 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
69 views

$X\sim\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. 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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28 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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Ratio of difference and sum of i.i.d. exponential random variables

I just noticed that for two i.i.d. exponential random variables $X$ and $Y$, the combination $\frac{X-Y}{X+Y}$ appears to be distributed uniformly on $[-1, 1]$ (ignoring the case $X=0, Y=0$ for the ...
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1answer
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Probability of sample distribution being different from population distribution

I have a population distribution I know is exponential, with mean $\mu_1$. I also have a sample of the population (of size $n$) that appears to be exponential with mean $\mu_2$. I have all the $n$ ...
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1answer
24 views

How Much Time Must the Shopkeeper Wait? — Exponential Distribution

In a store, the distance between customer arrivals follows an exponential distribution with a parameter of 8 minutes. The second seller starts his shift at 10:30 while the last customer entered the ...
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1answer
39 views

Does the distribution of $XY$ depend on $\theta$, when $X\sim\text{Exp}(\theta)$, $Y\sim\text{Exp}(1/\theta)$ and $X$ independent with $Y$?

Does the distribution of $XY$ depend on $\theta$, when $X\sim\text{Exp}(\theta)$, $Y\sim\text{Exp}(1/\theta)$ and $X$ independent with $Y$? I understand that from Wiki Parametrization 1 we have $XY$ ...
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1answer
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Pivotal quantity inference statistics of Exponential distribution?

Bus waiting times are distributed like this (they are independent) I know the average time is 8 minutes. I need to find the pivotal quantity of Theta parameter and after it of P. (P is the ...
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naive bayes classifier with Exponentially distributed likelihood with big parameter

Just for the practice of it, I'm trying to do a naive Bayes classifier for data which has exponential distribution for the likelihood function, i.e. $X_k=x|Y=1 \in Exp(\lambda_k)$ where $k = 1,..., p$ ...
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39 views

Gutenberg-Richter Recurrence Law: why is rate defined as probability of being exceeded?

According to Kramer (1996): Guttenberg and Richter gathered data for Southern California earthquakes over a period of many years and organized data according to the number of earthquakes that exceeded ...
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1answer
50 views

MLE based on bivariate data

Let $ x \sim Exp({\lambda}_{1}) , Y \sim Exp({\lambda}_{2})$ and are independent . We observe Z and W with Z = min(X, Y) and $W = \begin{cases} 1 &, if Z=X \\ 0 &, if Z = Y. \end{cases} $ Now ...
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Solution verification for an equation of exponentially distributed random variables

Given two i.i.d. random variables $X,Y$, such that $X\sim \exp(1), Y \sim \exp(1)$. I am looking for the probability $\Phi$. However, the analytical solution that I have got does not match with my ...
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Fat tails vs heavy tails

There is a lot of confusion about these two terms also in previous asked questions. I am interested in this part of the Wikipedia article (I am only considering right-tailed distributions): https://en....
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1answer
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Is Weibull distribution memoryless?

I googled and it seems not. Only exponential distribution is memoryless. Does anyone have an intuitive explanation why it is not? Thanks
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1answer
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Find joint density given conditional density

These are the steps I followed: $ \frac{f(y_1,y_2)}{f(y_1)} = f(y_2|y_1) $ $ f(y_1,y_2) = f(y_1).f(y_2|y_1) $ $ f(y_1,y_2) = 3e^{-3y_1}.\frac{1}{2y_1} = \frac{3}{2y_1} e^{-3y_1}$ Inorder to find ...
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How do I construct a symmetric, two sided confidence interval for a given statistic?

Let $X_1,...,X_n$ be a random sample of iid random variables, $X_i\sim Exp(\lambda),\lambda>0$. Consider the statistic $T_1(X_1,...,X_n,\lambda)=2\lambda n\bar{X_n}$. The task is to construct a ...
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Why doesn't $(e^{A})^{-1} = e^{-A}$ hold for a symmetric matrix in Python?

$e^A$ is just the $A$ matrix with all of its elements exponentiated, called a matrix exponential. It follows that the inverse $(e^{A})^{-1} = e^{-A}$ for square matrices, although I could find nothing ...
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Probability of X greater than Y with different types of random variable [duplicate]

My problem is the following: I have 2 random variables $X \sim Gamma(2,\mu_2)$ and $Y \sim Exp(\mu_1)$. I have to compute $P(X > Y)$. How can I do that ? Thank you
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Deriving the PDF of an exponentially modified Gaussian RV

For a random variable $Z = X + Y$, where $X$ is an exponential RV with $λ = 1$ and $Y$ is a Gaussian (Normal) random variable with mean $μ$ and standard deviation $σ$, how could we derive the ...
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1answer
37 views

MLE of (reciprocal) Exponential Distribution

I have a question in my homework - I worked out that the Likelihood is $ \frac{1}{\theta^n} e^{-\frac{1}{\theta}\sum y_i}$ The log likelihood is $-n\ln(\theta) - \frac{1}{\theta}\sum y_i$ ...
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Probability of being still in the system in a queue system

I have one queue with two servers $S_1$ and $S_2$.The serving times are modeled $\sim exp(\mu_1)$ and $\sim exp(\mu_2)$ respectively. The first server is free while the second has two clients, $A$ ...
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rpart with survival data: what do the reported rates mean?

This is essentially the same question as this one but the accepted answer doesn't answer it. I made data consisting of three groups, each following an exponential distribution with rates 1, 3 and 6 ...
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1answer
55 views

Exponential distribution and gamma prior

I want to use Bayesian conjugate to update my prior. Let's say I model bus arrivals by Exponential distribution with lamba=0.5. It means on average I will wait for <...
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Poisson process - time of arrival of a client, given that one client arrived at first interval

I have the following situation: I have a Poisson Process with $λ=7$ (seven customers / hour). This process describes the arrival of customers in a store. The store is open from 9:00 AM to 19:00 PM. My ...
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1answer
24 views

Probability of service in a queue theory problem with exponential random variable

I have one queue with two servers $S_1$ and $S_2$.The serving times are modeled $\sim exp(\mu_1)$ and $\sim exp(\mu_2)$ respectively. The first server is free while the second has two clients, $A$ ...
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1answer
24 views

Exponential Distribution problem with many items to be tested

I have the next exercise: The duration time X, in months, of a type of electrical resistance has the next probability density function: Then if 10 electrical resistors are tested, what is the ...
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2answers
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How to find a good estimator for $\lambda$ in exponential distibution?

I have an Exponential distribution with $\lambda$ as a parameter. How can I find a good estimator for lambda?
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Fitting exponential curves to data using maximum likelihood function

I have a data-set and I'm exploring potential exponential fits to the data using eye-balled estimations and estimations via maximum likelihood methods. I'm finding a huge discrepancy in one of the ...
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3answers
89 views

Binomial and exponential distribution

If I am given $N$ is from a binomial distribution with the parameter $n,p$ and also $X$ is from exponential distribution with the parameter $\lambda$. Assume that $S_N=X_1+X_2+...+X_N$. How can I find ...
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1answer
92 views

Two sample test for exponential distribution with only two observations

Suppose we have two independent random variables $X_1 \sim \exp(\lambda_1)$ and $X_2 \sim \exp(\lambda_2)$ . Now, we are given just one observation each from the two distributions above, say $S_1$ and ...
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1answer
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How to get coefficients from exponential model [closed]

I have data $y$ (contain some NA) and $x$ and i want to fitted an exponential model that follow the function: $$y=A \exp(B\cdot x)+C$$ What function in R that can estimate the value of $A$, $B$, and $...
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How to test if a process is poisson, if there are a lot of zeros

I have foot traffic data for how many people entered a building for every hour, for several days. This SOUNDS like it should follow a poisson process. Problem: I need to statistically confirm that my ...
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Statistical power of the Lilliefors test: how to compare results for samples of different sizes?

The power of the Lilliefors test (LT) strongly depends on the sample size. I need to apply LT to a discrete distribution to find the minimum value m_min above which it is exponential. I need to do it ...
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0answers
31 views

Laplace transform of sum of $N$ IID random variables where $N$ is itself random

Let $\{Y_i\}$ be a sequence of IID random variables so that $Y_i \sim Exp(\lambda)$ or equivalently $Y_i \sim CPH_1(1, -\lambda)$ (continuous phase-type distribution). Let $N$ be a discrete random ...
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1answer
40 views

What does it mean to be memory less for a random variable?

I have seen in class the concept of a memory less random variable, I can write it formally as : $$ P (X > t + s | X > s) = P(X>t) $$ And I understood the mathematical demonstration for the ...
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1answer
62 views

Generating samples on an exponential distribution

I am trying to generate a synthetic earthquake database where the number of events ($N$) with magnitude ($M$) in the range $[M, M+\delta_M]$ follows: $\log_{10}(N) = a - bM$ where $a$ and $b$ are ...
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Let $X_1,…,X_n\sim\text{Exp}(\beta)$. Find the moment generating function of $X_i$. Prove that $\sum_{i-1}^{n}X_i \sim \text{Gamma}(n,\beta).$

The following is a problem from Wasserman's All of Statistics Problem Let $X_1,...,X_n\sim\text{Exp}(\beta)$. Find the moment generating function of $X_i$. Prove that $\sum_{i-1}^{n}X_i \sim \text{...
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How does link function work in GLM?

I have several questions regarding the link function of generalized linear regression. I know how link function changes range of the distribution function's mean to the complete real line. But is that ...
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1answer
141 views

Conditional Distribution of $X-Y$ given $X > Y$ when $X$ follows exponential distribution [closed]

Suppose $X$ follows exponential distribution with a positive parameter $\lambda$ and $Y$ is a positive continuous random variable, independent of $X$. Then what is the conditional distribution of $X-Y$...
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2answers
356 views

Computing variance from moment generating function of exponential distribution

I'm wondering how to get variance of exp. distribution from the raw variance computed using the moment generating function. Here's my line of reasoning: PDF of Exponential distriution is $$ p_X(x) = \...

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