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Questions tagged [exponential]

A distribution describing the time between events in a Poisson process; a continuous analogue of the geometric distribution.

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Class frequency comparison (exponential distribution)

I have two different sources of data and I would like to check if their class distributions are similar and how much. The first dataset ($D_1$) has 2M samples and the second ($D_2$) 6M. When I ...
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Can I use the exponential distribution to model data with some negative values?

my data ranges between x=(-10, 60). X represents energy savings after adopting an energy efficient product, where most individuals save energy (positive x) but a few use more energy (such as through ...
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Linearization of multiplicative and exponential regression models and their OLS estimation

So far I have been under the impression that you can "linearize" multiplicative models of the form (1) $y=\alpha * \beta_1x_1 * \beta_2x_2 * \beta_3x_3 $ and exponential models of the form (2) $y=\...
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Distribution of a function of exponential and uniform random variables? [closed]

Consider the following independent random variables: $$\begin{equation} \begin{aligned} A &\sim \text{Exp}(\lambda_1), \\[6pt] C &\sim \text{Exp}(\lambda_2), \\[6pt] B &\sim \text{U}(0,1),...
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69 views

Confidence interval for $\sigma^2$

I started with any distribution and underwent the CLT on $\sqrt{n}(\widehat{\sigma}^2 - \sigma^2)$ where $$ \widehat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n (X_i - \mu)^2 $$ is a sample mean of $\sigma^2$...
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Exponential Family Representation: Dumb question on scale parameter and whether it went to Hawaii

So going over the Hastie Tibshirani paper on GAM - it points to equation 11 as the exponential family density - but with two parameters - theta (natural parameter) and phi (scale). https://...
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Find joint distribution for two different cases Kruskal Wallis

I'm a bit stuck with my homework in a subject called "Non-parametric Statistics". The task is related to Kruskal-Wallis test. The task is as follows: Let's look at the comparison of 3 independent ...
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35 views

The probability distribution of waiting time until two exponentially distributed events with different parameters both occur

I am working on a problem related to the waiting time until a parking garage is empty. We are given that the cars independently spend an exponential distributed time in the parking garage, with ...
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Joint reliability model

I have a following problem. There is a device where I expect exponential reliability model with parameter $\lambda$. So the probability of failure is $$P[T\ge t] = e^{-\lambda t},~t\ge 0$$. If the ...
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1answer
96 views

What should my critical region look like in this LR test for shifted exponential distribution with pdf $e^{-(x-\theta)}\mathbf1_{x>\theta}$?

I have a small confusion over describing the cutoff point for the critical region in a likelihood ratio test when the null hypothesis is composite. Take this exercise in particular: Let $(X_1,X_2,\...
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1answer
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probability that matrix $2\times2$ of Random variables is Invertible

Let $X_1, X_2, X_3, X_4$ to be Variables, and let $A$ be the following matrix: $$ \left[\begin{matrix} X_1 & X_2\\ X_3 & X_4 \end{matrix}\right] $$ assume that $X_1, X_2, X_3, X_4$ are ...
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Applying Expectation Maximization Algorithm to Mixture of Exponential Distributions

I am new to EM algorithm and I'm dealing with the following question of a Mixture of Exponential Distributions. Suppose that the time from when a machine is manufactured to when it fails is ...
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Exponential heteroskedasticity

Assume that the form of heteroskedasticity is exponential such that $u_{i} = e^{0.5X_{i}\gamma}\upsilon_{i}$ where it is assumed that $\upsilon_{i} \mid X_{i} \sim N\left(0,1\right)$ and $X_{i}$ ...
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Comparing two sets of exponential data $T(t)=−e^{−kt}$

I have two sets of exponential data (temperature measurements) of the form: $T(t)=−e^{−kt}$. k is a constant that determines the rate of temperature change. 1 temperature measurement was taken every ...
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45 views

Differences between exponential distribution with very different n sizes

I need to test for the differences between three groups of observations (grouped along $x$-axis), which appear to follow an exponential distribution along the $y$-axis dimension (see example fig.). ...
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27 views

Does exponential waiting time for an event imply that the event is Poisson-process?

Say I have a process, $\{N_t : t \ge 0\}$, which denotes the number of the event that occurred until the time $t$. And let me define $W = \min \{t : N_t = 1\}$ which is denotes the time until the ...
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Marginal Distribution of Exponential Mixture Model

I am currently trying to marginalize over the scale parameter in a mixture distribution of exponential pdfs, but I do not trust my result. Let me show you my steps: Probability Density Function The ...
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1answer
57 views

Survival time problem exponential with gamma prior

The survival times, in days, of patients diagnosed with a severe form of a terminal illness are thought to be well modelled by an exponential($\theta$) distribution. We observe the survival times ...
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Using fitted lagged variable as dependent variable in a new regression?

Suppose I have regression like this; $y[t] = a * exp(b*y[t-1])$ From this regression, I get; $\bar y = y - residuals$ What happens if I regress a new regression like this? $y[t] = c * exp(d*ybar[t-...
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1answer
75 views

Memoryless property of exponential

Let $X$ be an exponential random variable. I have been asked to evaluate whether each of the following is true or false and am seeking some insight about a solution that was offered. $(a) E(X^2 |X>...
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Test if long-term growth rates are constant

For example, consider U.S. economic GDP. This appears to be growing exponentially over time. Clearly, annual GDP growth is sometimes high and sometimes low. But say that one person claims that, over ...
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First moments of GBM-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu dt+ \sigma dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that $W_t \sim EMG^-(0,1,\lambda)$. Where ...
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2answers
52 views

Finding pdf with more than one random variable

I am stuck with a question doing one of my stats tutorial and question is as follows: Suppose X and Y are two independent exponential random variables with parameter $\theta$, i.e. their joint ...
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What is the limit of this expression?

If $\det(\Lambda_0) \to 0$, what does $$ \exp\left(-\frac{1}{2}\text{trace}\left(\Lambda_0 \Sigma^{-1}\right)\right)\det\left(\Lambda_0\right)^{-1/2} $$ approach? I was trying to answer the ...
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Hypothesis test for machine failure rates

Let's say I have one collection of machines (made by manufacturer 1). These machines run over a few days and fail at a certain rate. Let's say they run for a total of $h_1$ machine hours and the ...
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28 views

Probability of stale cache hits w/ exponentially distributed read and write inter-arrival times

I am trying a create a model for a very simple system which consists of a server and 2 clients: R and W. The server stores a piece of data D and hands out leases for caching D on the clients. The ...
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1answer
168 views

Discrete Fourier transform of an exponential decay

I have a vector with an exponential decay signal, using Numpy: t=np.arange(128) a=0.1 decay=np.exp(-a*t) I would like to compute the discrete Fourier transform (...
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1answer
37 views

What probability distribution can be described as y ~ X * Exp(1) [closed]

The green dots show the product of X and the exponential distribution with rate 1. It looks like it should be some simple probability distribution but I can't figure out what it is! Can you tell me ...
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2answers
107 views

Batch loss of objective function contains exp becomes nan

I am trying to solve a survival analysis problem, where all data are either left-censoring or right-censoring. I use an objective function which contains the CDF of Gumbel distribution. I have $m$ ...
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exponential of Covariance of two variables in logs, cov(X,Y)=exp(X)*exp(Y)*cov[ln(X),ln(Y)] [duplicate]

I am stuck in calculating steady state in a model that has covariances in logs. I am wondering in general if the following accurate. cov(X,Y)=exp(X)*exp(Y)*cov[ln(X),ln(Y)] if that is accurate ...
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Is the truncated power law a heavy-tailed distribution?

A heavy-tailed distribution is often defined as a distribution with a tail that is not exponentially bounded. A truncated power law (or power law with exponential cut-off) is a distribution that ...
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1answer
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Censored regression methods for analyzing extreme end of a normally-distributed variable

I have a normally distributed continuous variable referring to an observed human behavior, and I'm interested in measuring or rather analyzing the extreme of this behavior, namely, the top 10% of the ...
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Prove that the interarrival times of a Poisson Process are all indipendent and identically distributed

{$N_t$} with $t\in \mathbb{R}$ is a Poisson process with intensity $\lambda \in \mathbb{R^+}$, so that 1) $N(0)=0$, 2) {N(t) is with indipendent increments and omogeneous increments and 3) $\...
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1answer
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Prior for mean unknown parameter

We know that there are two versions of exponential $\big(1\big)$ $exp(\lambda)$ with pdf $$f(x;\lambda)=\lambda e^{-\lambda x}$$ ,$\lambda>0,x>0$ and $\mathbb{E}[x]=\frac{1}{\lambda}$ for ...
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How to model exponential decay with seasonality

I am trying to model sales driven by marketing tactics with promotional deadlines in Excel. The overall trend of the sales driven by each marketing tactic decays exponentially following the launch ...
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1answer
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Interpretting the p-value when inverting the null hypothesis

I'm performing an analysis by fitting data points to an exponential distribution and testing the adequacy of this fit using the Kolmogorov-Smirnov test. The standard ks test null hypothesis: the data ...
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1answer
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How to calculate mortality rate or probability of death at time t for various parametric distributions, e.g. Weibull, exponential, log-normal

If I have lambda and gamma, can I estimate that the probability of death at time t, based on a Weibull distribution is: What are similar formulas for the exponential, Gompertz, log-normal, log-...
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1answer
103 views

Piecewise exponential model - how to fit

I'm trying to go through the following paper: https://projecteuclid.org/download/pdf_1/euclid.aos/1176345693 it covers the piece wise exponential model for modelling hazard rates in a non-parametric ...
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1answer
273 views

Exponential Smoothing & Seasonality

I am relatively new to stats and forecasting and was hoping to tap the wisdom of this community in regards to a question. One of my colleagues insists that exponential smoothing presumes seasonality. ...
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1answer
63 views

Normalize number according to exponential function

I haven't picked up a math book in a while so please be patient. I have a series of numbers ranging from zero to infinity (realistically up to 10 though) and I want to normalize these values to 1 to ...
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1answer
86 views

Linear regression and exponential distribution

I'm running a simple ordinal least squares regression. My dependent variable is normally distributed, while my independent is exponentially distributed. Will this be a problem and, if yes, is there a ...
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1answer
2k views

MAPE Value more than 100% [closed]

Im using Trend Analysis - Double Exponential Smoothing Plot (not seasonal and it has a trend testedly) to forecast the net amount of carrier switchers (using Mobile number portability) in the future ...
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37 views

Testing Parameter Restrictions in R

I've got duration data that I'm analyzing in R. The exponential and weibull distributions are both commonly used when constructing Accelerated Failure Time models for duration analysis/survival ...
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3answers
329 views

Why do we use the natural exponential in logistic regression?

I would like to intuitively understand the benefit of using the natural exponential in the sigmoid function used in logistic regression. Why should it have to be $e^x$ instead of, for example $2^x$?
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Summation of Series involving Exponential terms

I'm currently working on a problem, which involves Poisson-Binomial Distribution. https://en.wikipedia.org/wiki/Poisson_binomial_distribution . The Mean of PBD is given by $M=\sum_{i=1}^{n}p_i$ ....
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1answer
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Distances between random points in a hypercube and statistics of exponents

TL;DR: Why is $\text{avg}\left(|a-b|^k\right)=\frac{2}{(k+1)(k+2)}$? I.e. for $k=2$, as for finding average Euclidian distances, the result is $\frac{1}{6}$? I've been reading a book about "Corobs," ...
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341 views

Exponential double decay curve fitting in Excel

Please refer the attached image. I tried to fit the curve with an double decay exponential function in excel using GRG nonlinear solving method. Every time I run the solver, I get the result stating ...
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Finding $\mathrm{Var}(N)$ if $N=\inf\{n\ge1:\sum_{i=1}^nX_i>1\}$ where $X_i$'s are i.i.d Exponential variables

Suppose $X_1, X_2, X_3, \ldots, X_n$ be independent and identically distributed random variables having an exponential distribution with mean $\frac{1}{\lambda}$. If $S_n = X_1 + X_2 + \ldots + X_n$ ...
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Mixing Distributions

Let $X|\alpha$ have a single-parameter Pareto distribution with parameters $\theta$ and $\alpha$, where theta is a known and fixed constant and $\alpha$ has an exponential distribution with parameter $...
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1answer
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Relation between skew and memorylessness of exponential distribution

When we have a random variable that follows an exponential distribution, the mode is at 0, the probability $P(\frac{1}{\lambda}<X<\frac{1}{\lambda}+\epsilon)$ is lower than $P(\frac{1}{\lambda}-...