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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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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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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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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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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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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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
150 views

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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Name for a distribution between exponential and gamma?

The density $$f(s)\propto \frac{s}{s+\alpha}e^{-s},\quad s > 0$$ where $\alpha \ge 0$ is a parameter, lives between the exponential ($\alpha=0$) and $\Gamma(2,1)$ ($\alpha \to \infty$) ...
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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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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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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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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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1answer
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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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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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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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1answer
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Distributive property of probabilistic inequalities involving random variables on both sides

Can I break down $P(h \geq (A + B)$, given all $ A,B,h$ are all random variables. Will the following rule works? $$P[h \geq (A + B)] = P(h\geq A) + P(h\geq B)$$ Actually, in one of my mathematical ...
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Memoryless conditional expectation of shifted function exponential

Related to this, is the following valid: \begin{align} E[f(X-t) \mid X>t] = \int f(y-t) f_{X|X>t}(y) dy = \int f(x) f_{X|X>t}(x+t)dx = \int f(x) f_X(x) dx = E[f(X)] \end{align} where I make ...
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Question about solution: Poisson process & conditional expectation

Given the following problem: Alice shows up at an Athena cluster at time $0$ and spends her time exclusively in typing emails. The times that her emails are sent are a Poisson process with rate $\...
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How to find exponential lambda parameters to maximise n parts of series of events likelihood?

I have a series of events. I think there are n periods that make up this series. (I do not know the bounds of the periods). I assume that the delay between the events of each of these periods follows ...
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Can I use z-scores with an exponential distribution? Or is there another test statistic for these types of distributions?

I have an exponential distribution for a population. $\theta = \mu = \sigma$ is known. Sample size $n$ is known. I need to find $"𝑃(𝑎<𝑋¯<𝑏)"$ for a random sample with size $n$. I think I am ...
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Showing bias of MLE for exponential distribution is $\frac{\lambda}{n-1}$

I want to show that the bias of $\hat \lambda = \frac{N}{\sum\limits_{i=1}^N x_i}$ is $\lambda/(n-1)$. There's a good chance that I'm too mathematically illiterate to understand the answer here ...
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distribution of quantiles, max, and min in normally and exponentially distributed data

I am wondering whether the distribution of different parameters (quantiles, min, max) of a dataset in different distribution (like normal and exponential distribution) follows the distribution of the ...
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What is the distribution of the k-th event in the Poisson process? [duplicate]

Assuming we have a Poisson process of density $\lambda$ I'm trying to find a distribution of a random variable $\tau_k$ - the time when k-th event has happened. E.g. in case of $\tau_1$ it is an ...
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Why are Poisson distribution and Exponential distribution special case of Gamma distribution?

I am aware that Gamma distribution is used as a conjugate prior distribution for various types of rate parameters such as in Poisson distribution and Exponential distribution. People say that ...
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How close to being memoryless can you make a distribution with bounded support?

Related to Exponential-like distribution with support [0,1] I wondered just how close to memorylessness a continuous distribution with bounded support can get. For a continuous variable to be ...
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Simulate Non Homogeneous Poisson Process

I've some problems with simulating arrival times following a non-homogeneous Poisson process. I am using the following arrival rate function: $\begin{equation*} \label{eq:lambda} \lambda(t) = \left\{...
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1answer
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Log-likelihood of a exponential distribution

I have an exercise that I don't quite understand: The life of 100 lamps has been measured. Each lamp has been used with a intensity between 0 and 1, where 0 is off and 1 is the maximum intensity. It ...
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Why does the distribution of the exponential random variable change to uniform distribution in this case?

I came across this very interesting question in a forum: If both X and Y are independent and exponentially distributed with parameter $\lambda$, find $E[X^2|X+Y]$ Someone gave the solution and ...
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What are NBUE ( New Better Than Used in Expectation) random variables?

I came across this term while reading a research paper but could not make any sense out of the information therein. Can someone please shed some light on what exactly does this mean ? The paper that ...
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What is the PDF describing the minimum distance in time between any event and the remaining N-1 events? (for a “process”-making events at fixed rate)

Previously I asked the following question, which is "What is the PDF for the minimum difference between a random number and a set of random numbers?" I want to translate what I learned from ...
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Wikipedia Proof About Minimum of Exponential Random Variables

In Wikipedia, for independent exponentially distributed random variables $X_1, \cdots ,X_n$ with rate parameters $\lambda_1, \cdots ,\lambda_n$, The probability $P(I=k)$ where $I=\textrm{argmin }_{i\...
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$M/M^B/1$ Burke's theorem : what is the distribution of the output batch interarrival times?

Setup: Take an M/M/1 queue: the inputs arrive according to a Poisson process at rate $\lambda$, the service time per item is distributed exponentially with mean $1/\mu$, $\mu > \lambda$ the ...
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How to generate exponentially distributed time period in a stateless manner

I am doing an objected based computer simulation, with some of the objects having a lifetime that persist for an exponentially distributed time. One way to do this is ...
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Is the question requiring the use of a gamma or exponential distribution? [closed]

Incoming telephone calls to an operator are assumed to be a Poisson process with parameter $\lambda$. Find the density function of the length of time for $n$ calls to be received, and find the mean ...
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Poisson Distribution with Exponential Parameter

If we have $X(k)\sim Pois(2k)$ and $Y \sim Exp(15)$ and $Z=X(5Y)$. How can we determine $E(Z)$, $Var(Z)$ and $P(Z = z)$. So far I'm thinking $$\begin{align*} E(Z) &= E(X(5Y)) \\ &= E(Pois(10Y))...
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What distribution could the data stem from?

From checking the histogram of the distribution I had the intuition that the data could follow a Poisson or an exponential distribtuion. However, Lilliefors test that the data is exponentially ...
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Exponential vs Gaussian Distribution in time problems

I'm wondering what about exponential distribution makes them better suited for time problems than gaussian distribution. For example, if I know that on average it takes the pizza delivery 20 minutes ...
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1answer
202 views

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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Kendalls Tau, for Exponential marginal Distribution

My Task is the following! $X_1 \sim Ex(2)$ and $X_2 \sim Ex(1/2)$ find a distribution so that $\rho_{\tau}=-0.85$. I have a little problem finding this distribution. Has anyone a clue what copula I ...
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How to find ks test statistic using the given maximum likelihood estimator values and a sorted data in R?

The random variable Y is said to have a two-parameter APE distribution denoted by APE(α, λ), with the shape and scale parameters as α > 0 and λ > 0, respectively, if the PDF of Y for y > 0 is ...
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Using chisq.test() to compare two exponential distribution variables in R

I want to use the chi-square test to judge if the variable X follows the exponential distribution. My plan is: 1] generate the X 2] using the rexp() to generate an exponential distribution vector 3] ...
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Suppose that X and Y are independent exponential random variables each with mean 1 [closed]

Suppose that $X$ and $Y$ are independent exponential random variables each with mean $1.$ What is $P\!\left(Y > X^2\right)?$
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Confidence interval for the maximum likelihood estimate of the minimum of a left truncated exponential distribution

I am currently working on a problem in which I have observations $y_{i}$ that are distributed, $y_{i} \sim \textrm{Exponential}(\beta = ax_{i})\cdot T[b, \infty)$ where, $\beta$ is the rate parameter ...
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conditional distribution in coin tossing problem

Let $X$ and $Y$ be exponential random variables with parameters 1 and 2. A coin has probability of getting heads as $p$ and probability of getting tails as $1-p$. Let $Z$ be another random variable ...
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What is the distribution of gap lengths in a Poisson process?

In a Poisson process with a finite period (and a known long-term-average event rate), what is the distribution of gap lengths between events? The number of events within a fixed period will be given ...
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Extracting maximum (x,y) values and initial slope of non-linear fitted glm curves

I have plotted 8 curves using a log-link Gaussian model, $y=ax \times exp(bx)+ϵ$, for my data. I am not sure how to extract the $max (x,y)$ from each of the fitted curves using the equation, of which ...