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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Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?

Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x}, \;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda e^{-\lambda x},\;\; y>0.$ Find the probability ...
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When to use weibull vs lognormal vs loglogistic for survival analysis regression?

When to use weibull vs lognormal vs loglogistic vs gamma distribution vs exponential when conducting survival analysis with survival analysis regression?
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Is there a model to predict mortality rate of an ongoing epidemic?

Following the recent Coronavirus outbreak, various mortality rates have been published. Most of these are simply the ratio between the deaths and the total verified cases, which is not a very ...
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Maximum Likelihood Estimator with exponential noise

So I need a little help with this please. I'm given N measurements of a signal $Y_{i} = A + v_{i}, i = 1,...,N$, where $v_{i}$ is measurement noise with the exponential pdf $f_{v}(v) = e^{-v}, v \geq ...
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Estimating a distribution from sums of samples

I'm trying to figure out the parameters of a distribution from real data, but I only get their sums and counts. For either exponential or normal distributions. So, I'll get the sum of 27 samples, ...
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What is the PDF for the minimum difference between a random number and a set of random numbers

I have a list (lets call it $ \{L_N\} $) of N random numbers $R\in(0,1)$ (chosen from a uniform distribution). Next, I roll another random number from the same distribution (let's call this number "b")...
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Q-Exponential distribution in Numpy

I am studying a phenomena, which I know is characterised by the q-exponential distribution, with parameters $c=1$, $q = 1.355$ and $b = 0.524$ in the equation $$e_{q,b,c}(t) = c(1 + b(q-1)t)^{\frac{1}{...
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1answer
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Exponential distribution with mean 1/lambda [duplicate]

Let x1,x2,x3,.......xn be independent and identically distributed random variables from exponential distribution with mean 1/Lambda. Let Sn=x1+x2+......+xn and N=infimum(n>=1:Sn>1) then what is the ...
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MLE $\hat{h(\mu)} = h(\hat{\mu})$ of $h(\mu) = var(Y_1) = \mu^2$

Question: Suppose Y1, · · · , Yn follows an Exponential distribution with $\lambda = \frac{1}{\mu}$. Derive the MLE $\hat{h(\mu)} = h(\hat{µ})$ of $h(µ) = var(Y_1) = µ^2$, and show that $h(\mu)$ is ...
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Convolutional roots of mixture of exponential distributions

In the reference book on infinite divisibility and generalised gamma convolution, BONDESSON, Lennart. Generalized gamma convolutions and related classes of distributions and densities. Springer ...
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How do I calculate Confidence Interval for Gamma Distributed Pivotal Quantity?

I'm studying confidence intervals and then I came across the following problem: It's said that a random variable X has Skewed Exponencial Distribution with parameters $\alpha >0$ and $v \in \...
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problem in understand exponential PDF [duplicate]

I'm studying a paper called "Optimization based on bacterial chemotaxis". As it can be understood from its name, it has proposed an optimization algorithm based on the reaction of a bacterium toward ...
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1answer
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Question about an exponential Bernoulli distribution

Mohie El-Din and Amein (2011) define a distribution in formula (1.2) which they call the exponential Bernoulli distribution (EBD). The distribution has the following form: $$\displaystyle f \left(t \...
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a question about method of moment estimator

I have a question about method of moment estimator. Say I have a IID sample $X_1, X_2, ..., X_n$ from an exponential distribution $Exp(\theta$), say I want to find the method of moment estimator of $...
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Can we consider arrival of pest outbreaks reports as Poisson Process?

I have a scenario where given a geographical region, farmers within that region can generate pest outbreak reports (like twitter messages) using a mobile phone. Also, a particular pest can spread to ...
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1answer
72 views

inverse of an exponential distribution

I have a question regarding this. Say I have $X_1, ..., X_n$ be random sample from an exponential distribution i.e. $Exp(\theta)$, and let $\gamma = \theta^2$. Let denote $\gamma^{mme}$ as the ...
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1answer
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Bivariate exponential distribution $(S, T)$ with controllable correlation and $S\leq T$

I am trying to define a bivariate exponential distribution $(S, T)$ with marginals $S\sim\mathrm{Exp}(\lambda_S)$ and $T\sim\mathrm{Exp}(\lambda_T)$ for $\lambda_S > \lambda_T$. I would like the ...
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2answers
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Finding the probability of survival of an insurance company

I was given as a homework exercise the following problem: however, I came into a disagreement with one of my classmates. Given that the solution is not shown, I was wondering whether mine was correct....
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1answer
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Hazard rate with exponential distribution

Hi I was trying to understand hazard rate and got stuck in the middle. Any suggestions are welcome. Below is the problem. Consider the following distribution for the duration of an unemployment spell ...
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Birth and death process, and calculating waiting time using Little's law

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible ...
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1answer
22 views

Finding limiting probability for continuous-time Markov chain

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible ...
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Incorporate sales forecast uncertainty into product stock availability simulation

I work in a e-commerce and I’m estimating availability of our catalog products. The process (in its simplified form) looks the following. A customer order comes in and we fulfill it if we have stock ...
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1answer
56 views

Find lambda for exponential distribution

I'm trying to find the function which allows me to find lambda if: $$y = 1-e^{-\lambda x}\,.$$ I tried doing this by: \begin{eqnarray} y-1 &=& -e^{-\lambda x}\\ e^{-\lambda x} &=& -...
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Probability of a 500 year flood occuring in the next 100 years - comparison of approaches

I'm looking at this problem A $500$-year flood is one that occurs once in every $500$ years. a) What is the probability of having at least $3$ such floods in $500$ years? b) What is the ...
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Suppose $\frac{1}{Y} \sim Exponential(\lambda)$, what is the pdf of Y? [closed]

I am actually attempting to answer the following: Suppose $\Pr(X|Y=y) \sim \Gamma(a,y)$ and that $\frac{1}{Y} \sim Exponential(\lambda)$ What is $\mathbb{E}(X)$? However, I know that this is a ...
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Methodology to distinguish single and double exponential distribution

I have a set of n data points $x = [x_1, x_2,...,x_n]$ that follow an exponential distribution or a double exponential distribution (two different $\lambda$ parameters). In order to check which ...
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2answers
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How does an observation condition the next one, if the numbers are exp. distributued with uknown average?

We have a process that generates exponentially distributed random numbers, i.e., $P(X=x) = \lambda e^{-\lambda x}$. However, we don't know the value of $\lambda$. We observe the first realization with ...
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Testing for Constant Hazard Function

My colleagues and I have some unemployment data, and we'd like to test the hypothesis that it's drawn from a memoryless distribution (= constant hazard function). Is there a standard test for this? (...
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1answer
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How to change exponential distribution into Normal distribution? [closed]

We have random data, which is exponentially distributed. Data = exp($\lambda$), where $\lambda$ = 0.5. If it is possible to change exponential distribution into the normal distribution. Then what ...
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Can the interarrival times of a continuous time markov chain be distributed with 2 parameter (scale,location) exponential distributions?

I'm trying to model data with a time-homogenous CTMC with a number of states with corresponding constant transition rates $\lambda_{i}$ when I notice that much of the transition times from one state ...
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Product of normal random variables and exponential random variables is exponential random variables?

How we can prove analytically, the Product of normal random variables and exponential random variables is exponential random variables?
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What is distribution of $\sin(x)$? If x is exponential distribution

I am trying to find analytically the distribution of $\sin(x)$, If x belongs to an exponential distribution,
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Probability of 100,000 Computer parts, if one computer part lasts more than seven years is $0.4966$

If the length of time the computer part lasts is exponentially distributed with mean value is $10$. So, for the exponential distribution, we can find the probability of one computer parts. $$p(x>...
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1answer
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Show that $nX_{(1)}$ is not consistent

Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
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336 views

Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
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451 views

Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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How to compare two statistical distributions with unavailability measurements?

I have two different measuring instruments to evaulate if an electronic device is working or not. These instruments provide a working/not-working reading each day and at the end of the month I compute ...
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2answers
109 views

Hypothesis test for composite null hypothesis of exponential parameter

I'm having trouble defining the reject region based on the generalized likelihood ratio test. This is from a question of past exam I'm self-studying and still have the doubt, given that I got it wrong....
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1answer
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Re-scaling of exponentially distributed random numbers

I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$ (for simplicity, $N=1$). I found that the generated numbers are initially exponentially ...
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How to show the variance of the inter-arrival time of a Cox process driven by a Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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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 ...
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1answer
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Mann-Whitney-U for exponential distributions?

I have two distributions, that look like the following: They appear to be exponential and have different sample sizes (369 vs 60). I would like to do some hypothesis testing. I know that I can use ...
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1answer
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Questions regarding this derivation of the Poisson Distribution from exponential densities

On page 217 - 218 of the pdf of this book, the author derives the Poisson Distribution using gamma and exponential densities. The author defines $S_n$ to be the sum of a sequence of independent ...
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If there are covariant variables in the exponential distribution, how do I draw the QQ plot?

Observation Data X=x[1],... x[n]; covariant variables are eta=eta[1],...,eta[n]. The parameters of exponential distribution are composed of covariant variables(eta) and lambda. So, X[i]<-rexp(eta[i]...
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To what distribution it's similar? Looks like an exponential but it's not

To what distribution it's similar? Looks like an exponential but it's not. It's seems to have a property, that if I zoom it (xlim) then each time it has the same ...
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Can Kernel Density Estimation estimate an Exponential Distribution?

Can Kernel Density Estimation estimate an Exponential Distribution? I tried to performed to make experiments with various kernels like: "gaussian" and "exponential", but performance seems to be very ...
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How do we build a confidence interval for the parameter of the exponential distribution?

EDIT Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\text{Exp}(\theta)$, where $\theta$ is not known. Precisely, $f(x|\theta) = (1/\theta)\exp(-x/\theta)$ Describe ...
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1answer
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Why can't I write $P(X>5|X>1) = P(X>5)$? [duplicate]

I have a confusion with the memorylessness property of exponential distribution. If exponential distribution is memoryless (i.e. the past has no bearing on its future behavior), why can't I write $P(...
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1answer
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exponential RV til bus arrives

Suppose that you are waiting at a bus stop. The waiting time until a bus arrives is $T$ where $T$ is an exponentially distributed random variable with parameter $λ$ i.e. $P(T≤t)=1−e^{−λt}, ∀t≥0$. (a) ...
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1answer
68 views

Geometric distribution described with rate parameter

I don't understand this sentence from this paper (around equation $5$): The function $H(\tau)$ is the hazard function. $H(\tau) = \frac{P_{\text{gap}}(g = \tau)}{\sum_{t=\tau}^{\infty} P_{\...