Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [exponential]

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

2
votes
0answers
20 views

Maximum entropy probability distribution over non-negative support and finite mean?

I'm trying to derive which univariate probability distribution maximizes entropy, assuming finite mean $\mu$ and non-negative support $[0, \infty)$. I know that the answer is the exponential ...
0
votes
0answers
16 views

Why knowing information about something doesn't help when it comes to Exponential Distribution?

I've recently learned about Exponential Distribution and when it's appropriate to be used. There was the following example given. The duration of IC555 integrated circuits under normal operation ...
1
vote
1answer
54 views

Two-sided UMP test for exponential densities?

I'm struggling with a problem from Lehmann & Romano's book *Testing Statistical Hypothesis." Suppose $X_i$ is a random sample from $$f(x) = \frac{1}{b}e^{-(x-a)/b}\mathbf{1}_{x>a}$$ The ...
0
votes
0answers
27 views

Binomial distribution in exponential form

I am trying to find out if I am doing this right: I have started with: $f(y)=\binom{n}{y}(\frac{\mu}{n})^y(1-\frac{\mu}{n})^{n-y}$ This is what I get: Result: $exp[ylog(\frac{\mu}{n-\mu}) - nlog(1+e^...
0
votes
0answers
13 views

What statistical test is suitable for testing the goodness of fit for an exponential regression model?

I want to do a study about the risk factors of heart failure disease that will affect the survival time of a patient. My independent variables: age, smoking status, ejection fraction etc.., and my ...
0
votes
1answer
22 views

Statistical specification for a Regression with continuous finite range count like data

I am interested in explaining what kind of personal characteristics and work environment variables are associated with sickness absenteeism. My dependent variable is the total number of days a given ...
0
votes
0answers
41 views

Back transforming the intercept of double log normal distribution

I apologise if this is a duplicate, i couldn't find a thread that seem to be talking about the same thing. I have a dataset with a bunch of duration of varying lengths in seconds, i log-transformed ...
4
votes
2answers
285 views

Sum of two exponential series with equal means and variances

Assuming $A$ and $B$ are two non-negative real-valued random variables such that $\mathrm{E}(A)=\mathrm{E}(B)$ (equal means) $\mathrm{Var}(A)=\mathrm{Var}(B)<\epsilon$ (equal small variances) is ...
10
votes
1answer
163 views

Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

Suppose $(X,Y)$ has the pdf $$f_{\theta}(x,y)=e^{-(x/\theta+\theta y)}\mathbf1_{x>0,y>0}\quad,\,\theta>0$$ Density of the sample $(\mathbf X,\mathbf Y)=(X_i,Y_i)_{1\le i\le n}$ drawn from ...
1
vote
1answer
37 views

Designing hypothesis test around failure rates

I'm trying to devise a hypothesis test for failure rate data of machines. The gist is that there are some machines in a factory that run all the time. They fail from time to time and are promptly ...
0
votes
0answers
3 views

Normalise exponential numbers between 0 and 1 [migrated]

I'm creating a fractal visualisation. I want the colour per point to be based off the iteration final value, $f(z_{n})$, instead of the traditional: number of iterations before reaching a cut-off (...
1
vote
1answer
33 views

How can I make a Kolmogorov-Smirnov test to check if my data distribution is exponential?

I made a histogram of my data, and the fitting line, but from some reason the fitting line doesn't fit to my graph. How can I make it fit to it? How can I check if my data distribution is exponential?...
1
vote
0answers
26 views

If $X$ is from (canonical) exponential family, do we always know the distribution of $T(X)$?

Assume $X$ are generated by a distribution from exponential family, $$ f(X; \theta) = h(X)\exp\{\eta(\theta)T(X) - b(\theta)\}$$ After solving several exercises with various distribution functions, ...
0
votes
0answers
10 views

Manually calculating the survival probability S(t) from flexsurv exponential regression parameters in R

I am learning survival analysis for my research. My question is very basic. I created a simulated dataset as shown below: ...
0
votes
0answers
16 views

How to properly truncate exponential distribution to represent random memory-less arrivals?

Typically, Poisson and exponential distributions are used to represent random memory-less arrival processes. It has come to my attention, however, that a more realistic distribution is a truncated (or ...
1
vote
2answers
38 views

Exponential Distribution - Memoryless and examples

In several (introductory) statistics books we can see that they use an Exponential Distriubtion to model the time of failure of an electronic component. I understand that it got the appealing property ...
0
votes
1answer
23 views

Why is sample mean minus location parameter of shifted exponential gamma distributed?

My book says the following Suppose $X_{i} \sim$ iid $Exp(1,\eta)$ Where $Exp(\theta,\eta)$ is the shifted exponential ie has density $$\frac{1}{\theta}e^\frac{-(x-\eta)}{\theta}$$ for $x \ge \eta$ ...
0
votes
0answers
13 views

median survival derived from a mixture of exponential distributions

I recently got a very nice question... Let's say that we have one cohort, and we know the median survival of it. Then we split the cohort into two groups, and somehow we also know the median survival ...
0
votes
0answers
45 views

Transforming back after a log transformation with subtraction

I needed help with back transforming my data. My initial data was positively skewed so I had to log transform it, after which I did my statistical test. One of my regression test required for my ...
3
votes
2answers
138 views

95% Confidence interval of $\lambda$ for $X_1,…,X_n$ IID exponential with rate $\lambda$

I know how how to find the estimation of $\hat{\lambda}$ using the method of moments. I can take the first moment and equate it to the empirical to get, $E(X) = \frac{1}{\lambda} = \frac{\sum_{i=1}^{...
0
votes
0answers
11 views

Exponential Test

Given a collection of data points, without any other information I want to test for an exponential distribution. In a different case, I tested for normality using an Accord implementation of the ...
1
vote
1answer
37 views

Alternative heavy right tailed distribution to exponential distribution

I have data whose distribution resembles an exponential distribution, but the data has a heavier tail than the exponential distribution. I will be very glad for any recommendation of an alternative ...
0
votes
1answer
56 views

calculate quantile for exponential distribution

My distribution function is: y=a * exp(-b*x) What is the right way to calculate first Quartile? Right now I do like that: Q1=ln(4/3)/b Is that correct?
0
votes
0answers
13 views

How to model count data with decay

I'm trying to understand how I might model count data where there's diminishing marginal utility and a stochastic process. So, let's say we're modeling the number of "useful intelligence tips" given ...
0
votes
0answers
109 views

Bus arrival times and exponential distribution

I came across a question that is supposed to show us how the properties of the exponential distribution can be used. I know and have shown that $$P(X_i<min\{ X_1,\dots,X_n\})=\dfrac{\lambda_i}{\...
0
votes
1answer
74 views

how to draw 100 numbers following exponential distribution (erlang) in python by having k and CV and mean exponential distribution?

how to draw erlang distribution (two parameters k and cv? (k=1,5,10, cv=100%-45%-32%) ) in python by having k and CV and mean exponential distribution=1? I know numpy.random.exponential(scale=1.0, ...
3
votes
0answers
42 views

two independent Poisson Arrivals

I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$ Consider two ...
0
votes
0answers
116 views

Conditional expectation of an exponential variable

I have trouble with conditional probabilities, therefore I am wondering if the following derivation is correct: Both X and Y are exponential random variables, with $\lambda_x$ and $\lambda_y$ ...
2
votes
1answer
79 views

Finding variance of infimum of a set [duplicate]

Let $X_1, X_2, \cdots,X_n$ be independent and identically distributed random variables having an exponential distribution with mean $\frac{1}{\lambda}$. Let $S_n=X_1+X_2+\cdots+X_n$ and $N = \text{...
3
votes
2answers
73 views

Best regression model for data that shows exponential decay, but has and is bounded by zero

Fairly new to regression analyses. I have a data set for a clinical metric that is bounded in the range of [0-38] for a number of anonymized patients. I want to develop a regression model that ...
3
votes
1answer
124 views

UMAU confidence interval for $\theta$ in a shifted exponential distribution

Suppose $X_1,X_2,\ldots,X_n$ is a random sample drawn from the distribution $$f_{\theta}(x)=e^{-(x-\theta)}\mathbf1_{x>\theta}$$ It can be shown that there exist some $c_{\alpha}, d_{\alpha}$ ...
1
vote
1answer
57 views

What does it mean by “approach the performance of the Bayesian gold standard”?

It is a sentence in Dropout paper(Dropout: A Simple Way to Prevent Neural Networks from Overfitting). "This can sometimes be approximated quite well for simple or small models, but we would like to ...
0
votes
0answers
24 views

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 ...
0
votes
1answer
33 views

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 ...
2
votes
2answers
43 views

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=\...
-3
votes
1answer
76 views

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),...
1
vote
1answer
75 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 $\sigma^2$...
1
vote
0answers
47 views

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://...
0
votes
0answers
12 views

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 ...
2
votes
1answer
47 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 ...
6
votes
1answer
312 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,\...
2
votes
1answer
45 views

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 ...
1
vote
0answers
68 views

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 $X_{i}$ ...
0
votes
1answer
54 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.). ...
0
votes
0answers
40 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 ...
4
votes
1answer
63 views

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 ...
0
votes
1answer
133 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 ...
1
vote
0answers
20 views

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-...
3
votes
1answer
172 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>...
0
votes
0answers
8 views

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 ...