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

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Number of Exponential Summands in fixed interval is Poisson

What is the most clever way to prove that the number of independent exponential summands in a fixed interval is distributed as a Poisson random variable? I can do it one way, but I would like to know ...
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22 views

Determining if points out of control - non normal distribution

I am trying to create a process in which I can identify if a process is out of control. My idea was to do something similar to 6 sigma, where when a point is outside of the mean by +-3 sigma, then ...
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42 views

Understanding the CDF of the Exponential from the PDF?

I was trying to get the CDF of the exponential through the pdf. I know that the relationship between the pdf and the cdf is that the pdf is the derivative $ \lambda \exp(-\lambda x) $. But I don't ...
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67 views

If X~Exp(λ), what is the expected value of Y=X²?

I am trying to compute this using the integral definition of expected value but I don't think I am doing it right as I am getting a very hard integral that I can not solve. When computing ...
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1answer
13 views

How to specify exact rise-to-peak and decay-to-baseline times in an exponential function, generating a waveform?

I am attempting to model the fluorescent signal emitted by a fluorescent calcium indicator (lights up when there is calcium influx into a cell). According to [1], the following formula works as a ...
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20 views

Exponentiating dependent and independent variables in regression

I have a regression where I exponentiated both the independent and dependent variables. (I am referring to something basically like this: $\exp(\% \Delta y_i) = \beta \times \exp(x_i) + ...
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2answers
68 views

How do the means of $X^2$ and $X$ compare?

If $X$ has an exponential distribution with mean $\theta$, does $X^2$ have mean $\theta^2$? If not, how would I find the variance of $X^2$? I tried this: $$V(X^2) = E[X^4] - E[X^2]^2$$ But I'm ...
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1answer
62 views

Finding parameters of a given function

I have this following relation $$N = \dfrac{(113834700(3000-c)(1-e^{-(Xn)/(3000-c)})}{n}$$ I also have a set of values for $N$ and $X$, these are vectors. So, we have a scatter plot $N$ vs $X$. We ...
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1answer
34 views

What exactly is a “truncated” power law distribution?

This paper describes the data they analyzed as following a "truncated" power law distribution: To me, this just looks like they multiplied a power-law distribution ($\Delta r + \Delta ...
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13 views

Fitting to an Exponential Decay with Constant Background [duplicate]

I'm looking for simple way to fit to an exponential decay with a constant background, so something that looks like: $y=ae^{-bx}+c$ I have an idea of how to do it without the background. I would ...
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248 views

Regression with skewed data

Trying to calculate visit counts from demographics and service. The data is very skewed. Histograms: qq plots (left is log): ...
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39 views

Mixing exponential and linear regression with multiple predictors

This is the data set I am working on, trying to predict count (last column) : ...
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9 views

Combining two time-dependent statistical distributions

I've got a bit of an interesting statistical conundrum. I've got a time-dependent process that I've been able to model as the combination of two different distributions. Let's imagine that I'm ...
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1answer
74 views

Multilinear loss in Exponential-Uniform model

Let a prior $\pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta))$ and $f(x\mid\theta)=\theta e^{-\theta x}$. Taking the multilinear loss ...
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9 views

scaling exponential function to control maximum activation and time starting point

I would like to model a brain activation as a function of time as an exponential growth. Specifically, I would like to scale the function to have 3 parameters: (i) one parameter indexing the maximal ...
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23 views

Understand Exponential Moving Averages in Matlab

I'm unable to manually replicate the exponential moving average values that I see when using MATLAB's tsmovavg function. There seems to be some ambiguity online ...
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17 views

Lack of memory: $P(X_1 \geq (X_2 -t) + t | X_1 \geq t) = P(X_1 \geq X_2 - t)$?

Suppose $X_1$ and $X_2$ are independent random variables, where $X_1$ and $X_2 - t$ are exponentially distributed (fixed $t>0$) with rates $\lambda_1 > 0$ and $\lambda_2 > 0$, respectively. ...
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27 views

Comparison of different forecasting models

I have a time series data of 1000 points (in %) for each of the different machines. I tried different forecasting techniques to make a one step prediction. The goal is to find out one common ...
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39 views

insurance exponential

i would like to ask for help with following example, i know how to derive next steps that I am not even showing here, but cant derive the loglike function, thank you.
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23 views

Joint PDF of sums of exponentially RVs with common terms

Let: $\gamma_{c_1}=\gamma_1+\gamma_3+\gamma_4$, and $\gamma_{c_2}=\gamma_1+\gamma_2+\gamma_4+\gamma_5$, where $\gamma_i, i=1,...,5$ are i.i.d. exponentially distributed RVs: ...
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18 views

Distribution of lengths of randomly cut string [duplicate]

If I cut a string at random (uniformly distributed) positions, what is the distribution of lengths of the resulting fragments? From what I understand the lengths should be distributed as a negative ...
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1answer
50 views

MLE for Independent Exponentials

Here is the problem statement: Let $(X_i,Y_i)$ be a random sample from a distribution with pdf $$f(x,y;\theta)= > \frac{1}{\theta^3}\exp\left(\frac{-x}{\theta}-\frac{y}{\theta^2}\right)\qquad ...
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18 views

How to apply glm(generalized linear model) in this simple example?

We are given 1) Y = $(Y_1,Y_2,...,Y_n)^T$ ~ Exponential 2) E[Y] = $\mu$ = X$\beta$, where X $\in R^{nxr}$ and $\beta \in R^r$ My question is can we apply the glm in this case? The case where the ...
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24 views

conjugate prior for exponential distribution

If there is an exponential distribution $$p(x | \theta) = \theta\,e^{-x\theta}\mathbb{I}_{x>0}\, ,$$ what is a good conjugate prior? Also, will the posterior mean is a convex combination of prior ...
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1answer
52 views

How are mixed exponential distributions fitted?

A single exponential is defined as $f(x;\beta)= \begin{cases} \frac{1}{\beta}e^{\frac{-x}{\beta}} & x \ge 0\\ 0 & x<0 \end{cases}$ and the MLE of $\beta$ can be estimated from samples ...
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20 views

Combining Exponential and Bernoulli distributions

I am building a model of customer spending, and need some help to identify the best way of doing this. (1) I have an exponential distribution for each customer's spending (2) Bernoulli distribution ...
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10 views

Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space [duplicate]

Suppose that we are given an exponential distribution model with a pdf $f(x,\theta) = \theta^{-1}\exp(-x/\theta)$ with an iid sample $X_1, ..., X_n$, and we would like to test hypothesis $H_0 : \theta ...
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21 views

Significance of a exponential mean

A manufacturer of bulbs claim that they have a lifetime of 2000 burning hours. A sample of 100 tubes were taken at random and tested for burning. Mean life was found to be 1950 with SD of 150. Can ...
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1answer
45 views

finding the expectation of a particular distribution

I'm dealing with the exponential distribution with mu$\neq$ 0 and am trying to find some expectations. I'm stumped. can anyone take a look at it for me? here's my question. $x_1,...x_n$ are iid ...
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25 views

exponential distributions of area and volume?

This seems like it should be obvious, but... In my research, I am measuring 3 variables: length, area, and volume. Our length measurements have a nice, normal distribution. Area and volume both have ...
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1answer
25 views

Mean value of a (portion of an) exponential distribution satisfying a given inequality

I have some questions about exponential distributions. I have tried googling around but with no success. Imagine we have an HTTP server which serves Web pages in $x$ milliseconds, with a mean ...
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1answer
42 views

Is it correct to use correlation coefficient in this case?

I have an exponential random variable $\tau$. I define two random variables $T_A=\min(t_1,\tau)$ and $T_B=\min(t_2,\tau)$, where $t_1$ and $t_2$ are constants. What does it mean to compute ...
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29 views

Average waiting time in an exponential distribution

I have come across the following problem in one of my courses: Let's suppose a patient in a hospital is waiting to be picked by one of the two doctors available. The time till a new pacient arrives ...
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26 views

Is there a general expression for ancillary statistics in exponential families?

It is known that an i.i.d sample $X_1,\cdots,X_n$ from a scale family with c.d.f. $F(\frac{x}{\sigma})$ has $S(X)$ as an ancillary statistics if $S(X)$ depends on the sample only through ...
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64 views

correlation coefficient for exponential model - the perils of Excel

Editing a non-peer-reviewed paper I came across a Excel scatter-plot with an "exponential trend-line" drawn, AND the value of the correlation coefficient presented alongside (calculated for the ...
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48 views

Self-Starting Nonlinear Regression Model

In the following two nonlinear regression models, is it possible for the dependent variable (y) or the independent variable (x) to be negative? \begin{align} y &= φ_1 \exp[-\exp(φ_2)x] + φ_3 ...
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51 views

Exponential regression and focus on small values

I have a set of data with 3 numeric variables: X, Y and Z. I have access to data with 15 ...
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75 views

Exponential distribution question

Let X and Y be independent exponential random variables with respective rate $\lambda$ and $\mu$ let c be a positive constant. I want to find out a) $E[min(X,Y)|X>c]$ b)$E[min(X,Y)|X>Y+c]$ by ...
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62 views

What is the ratio distribution of a spacing and the sample mean?

Let $X_1,\dots,X_n$ be a sample of iid exponential random variables with mean $\beta$, and let $X_{(1)},\dots,X_{(n)}$ be the order statistics from this sample. Let $\bar X = \frac{1}{n}\sum_{i=1}^n ...
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1answer
73 views

Relationship between λ (or μ) in Poisson and Exponential distribution?

Is there any relationship between λ (or μ) in Poisson and Exponential distribution? In other words, if I know λ (or μ) for one of the distributions, is it possible to calculate the corresponding value ...
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35 views

Sufficient statistic for expectation of exponential family

True or false: Let X $\sim$ f, where f is element of an exponential family. Then, $\frac{\sum_{i=1}^n x_i}n$ is a sufficient statistic for $E(X)$. For either case, please provide the ...
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33 views

Survival Analysis - Improbable Exp(B)?

I'm currently conducting a survival analysis on my dataset. I have data on adoption of a new service and am using a Cox Proportional Hazard Model in order to account for right-censored observations. ...
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35 views

How can I evaluate the upper or lower bound for a function of random variables?

Let $\Gamma_1$,$\Gamma_2$ and $\Gamma_3$ be random variables defined as: \begin{align} \Gamma_1 &= \frac{XY}{X + Y + 1} \\[10pt] \Gamma_2 &= \frac{XY}{aX + bY + c} \\[10pt] \Gamma_3 &= ...
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98 views

Time to second failure of independent exponential lifetimes

This maybe an easy question, but as I am a beginner, I need help. Suppose that an electronic system contains $n$ similar components that function independently of each other and that are connected ...
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105 views

Related to exponential distribution

I am a beginner in statistics.(and probability theory too).I studied Exponential distribution and just started doing problems when I got stuck in the following one: Suppose that $X_1, . . . , X_n$ ...
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1answer
34 views

exponential likelihood with normal prior

If I have a likelihood function based on the exponential distribution with parameter $\lambda$ , why would a normal prior with a very large variance be inappropriate for $\lambda$?
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1answer
28 views

Exponential Regression with x outside of exponential

I am trying to do exponential regression by matrix notation, and I am trying to figure out to create my $\mathbf{X}$ matrix to fit my model. I know that I need to use a model function of the form $c_1 ...
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68 views

Parametric Competing Risk Survival Analysis - Exponential Distribution

I am familiar with the literature on nonparametric estimation of the cumulative incidence function under a competing risk model. I'm looking for the more basic solution under exponential ...
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19 views

Compute the probability of k events within time period during overvation period

The Erlang distribution is the distribution of the sum of k independent and identically distributed random variables each having an exponential distribution. Thus, it can be used to compute the ...
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1answer
88 views

Distribution of $\sum_{i=1}^d | \mathbf{u}^H \mathbf{F} \mathbf{v}_i |^2$ if $| \mathbf{u}^H \mathbf{F} \mathbf{v}_i |^2$ is exponentially distributed

Let $\mathbf{u}$ and $\mathbf{v}_i$ be an $M \times 1$ and $N \times 1$ vectors of unit norm, respectively. $\mathbf{u}$ is a column of a unitary matrix $\mathbf{U}$ and the $\mathbf{v}_i$ are ...