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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Calculate mean and quartiles for exponential distribution, which consists of a sum of two E-functions [closed]

How to calculate the mean, quartiles and variance for an exponential distribution with the following function: 17/99 e^(-0.5y) + 82/99 e^(-0,25y)
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2 answers
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Why doesn't R use Inverse Transform Sampling to sample from the Exponential Distribution?

I was reading this question about the algorithm that R uses to sample from the Exponential($\lambda$) distribution. It looks like R uses the Ahrens-Dieter algorithm to sample from the exponential ...
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Question about how to interpret certain data (am I correct in using the exponential distribution)?

Let's say my data has certain parameters $x_{1},...,x_{n}$ and there are two events, let them be $h_{1},h_{2}$. I'm considering interpreting the data via an exponential distribution because in this ...
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Why do simulated arrival times from a Poisson distribution seem to show periodicity?

I am experimenting with simulated arrival times drawn from a Poisson distribution. To construct the arrival times, I am randomly drawing inter-arrival times from the inverse CDF, which is ...
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modeling time between calls with exponential distribution

I've read that time between calls (in a call center) can be modeled with exponential distribution. My question is this: the shape of the exponential distribution has a decreasing nature. Suppose that ...
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Calculating the exponential growth rate against the standard deviation of the year coefficient

I have time-series abundance data for various locations. I would like to calculate the exponential growth rate for each location against the standard deviation of the year coefficient. My dataframe ...
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DataCamp exercise about distributions

I was studying some statistics in DataCamp and they assigned me this exercise that I can't solve. I tried speaking with people that know more statistics than me and we can't seem to agree in an answer....
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Difference between Power law distribution and Exponential distribution?

What is the difference between Power law distribution and Exponential distribution? They both look similar!
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Is log-rank parametric or non-parametric test, and why?

How is the log-rank test a "non-parametric test" according to wikipedia.org if one has to specify the parametric survival model for this test? We may run log-rank under the assumption of ...
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Statistical Inference Casella & Berger Exercise 7.11 [duplicate]

I'm self studying statistics using Casella & Berger Statistical Inference and I'm confused about a detail in solution to exercise 7.11. Here's the problem I'm try to solve: Let $X_1, ..., X_n$ be ...
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1 answer
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Probability of $A<B$ when $A$, $B$ are random variable with different distribution?

Preparing exams, I ran into the following problem: Edit: it shouldn't be represented as it was. Added the storkes. Let $A$, $B$ be two independent variables having probability distribution: $$ \...
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How to fit exponential and Poisson distributions to table of event times in R?

I have a vector of event dates where multiple events can occur on the same date. I have sorted these dates in chronological order and have also generated a table of times between events. Here is an ...
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Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable

Consider a random variable \begin{align} U= a \log (Z+b)-Z \end{align} where $a,b>0$ and $Z$ is an exponential random variable. Question: Can we find the variance of $U$? Things that I tried ...
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Sum of iid Exponential observations then subtracting the minimum of the observations [duplicate]

Consider $n$ iid $X_1,...,X_n \sim Exp(1)$. My goal is to find the density of $\sum (X_i - X_{(1)})$. My attempt If we write out the entire summation in order statistics, we get $X_{(1)}-X_{(1)} + X_{(...
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Exponential, Poisson or neither?

I have two variables in a marketing context: Advertiser spend per hour and conversions per hour. kernel density approximations of the underlying distributions look like below. Both distributions have ...
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Interpretation of drm parameter estimates and p-values for EXD.3 function in 'drc' package in R

I was wondering if someone could help me understand what the parameter estimates and p-values are saying in a three-parameter exponential decay function using the drm function in the 'drc' package in ...
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2 answers
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Goodness of fit for exponential distribution and large sample

I'm new to statistics and statistics are not my area of research so maybe my question is simple and answer is on the surface. In my research the empirical CDF for the data looks like exponential ...
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Exact/Approximate Confidence Interval of Parameter Ratio from two Samples of iid Exponential

Suppose 2 independent samples $X_1,...,X_n \sim Exp(\lambda_1)$ and $Y_1,...,Y_m \sim Exp(\lambda_2)$, and are iid within samples. I am thinking about how to make an exact confidence interval for $\...
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MLE of parameters for a difference of two Exponential IID

Suppose I have $X_1 \sim Exp(\theta_1)$ and $X_2\sim Exp(\theta_2)$. Then it is not difficult to show that $Y = X_1 - X_2$ will have density: $f_Y(y) = \frac{1}{\theta_1 + \theta_2}e^{-y/\theta_1}\...
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How to compute expectation of a exponentially distributed variable given the value of another variable?

I have 2 mutually independent random variables: $s$ is distributed exponentially with parameter $\lambda$: $s\sim F(\cdot|\lambda)$ $\epsilon_x$ is distributed exponentially with parameter $\chi$: $\...
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2 answers
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How to computed "truncated shifted exponential distribution"?

I have a research problem to solve. For regular exponential distribution, $$F(z|\lambda)=\begin{cases}0\;\;\;\;\text{if }z<0\\1-e^{-\lambda z}\;\;\;\;\text{ if }z\geq 0\end{cases}$$ with density $$...
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How are sums and differences of independent Exponential random variables distributed?

If we are given X1, X2, X3, X4 , all exponential random variables with the same mean λ and said that another random variable T = X1 + X2 + X3 - X4. Can we then say T basically has a Gamma ...
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Biasness of an estimator depends on whether you take expectation of the estimator or its inverse

(Please read until the end) know Consider two ways of writing the exponential distribution- (A) $\frac{1}{\beta} e^{-\frac{x}{\beta}}$ and (B) $\theta e^{-x\theta}$ If I estimate $\beta$ or $\theta$...
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Write the PDF of an exponential prior given E[$\theta$] = 2

I am reviewing old exercise solutions and the following info is given: Assume that the conjugate prior for θ (as a special case of the gamma distribution) is following the exponential distribution ...
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Find pdf of X+Y [duplicate]

Let X ∼ Exp(λ) and Y ∼ Exp(μ) be two independent exponential random variables, where λ, μ > 0. Find the probability density function of X + Y if λ ̸= μ. I have successfully find ans if λ = μ, but ...
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What are the operational links between exponential and weibull distribution

I recently read a very interesting article about competing risks that came with simulations. In these, the author defines a Weibull distributed variable w using an ...
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What is the name of the functions in exponential dispersion family?

If an exponential family is given by: $g(y|\theta) = exp\{\theta^TT(y)-A(\theta)\}h(y)$ then the functions $h(y)$, $A(\theta)$ and $T(y)$ are defined by names: $T(y)$ is a sufficient statistic $A(\...
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Calculating expected value

Came across an interesting problem. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it to the ...
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Correlation and Max and Min of Two Exponentials [duplicate]

Came across an interesting problem: Let X and Y be independent random variables such that both X and Y ∼ Exp(1). Define L = min(X,Y) and H = max(X,Y). What is ρ(L, H)? Why must the correlation should ...
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Generating multivariate random variable with normal and exponential marginals

I have a collection of data points of the form $[U, V, X, Y]$, where $U$ ~ $N(\mu_1, \sigma_1)$; $V$ ~ $N(\mu_2, \sigma_2)$; $X$ ~ $exp(\lambda_1)$; and $Y$ ~ $exp(\lambda_2)$, and I am looking to ...
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Unexpected distribution of ab-cd where a,b,c,d are independent and N(0,1) distributed

I have discovered an unexpected curiosity that if a,b,c,d are independent random variables and are N(0,1) distributed, then |ab-cd| is exponentially distributed Exp(1). It's easy to verify that ...
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Solving for the parameter of an exponential distribution

Suppose I have a random variable $X$ where $X$ follows an exponential distribution of the following form: $$f_X(x) = \frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$. I want to find the value of $\lambda$ ...
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Exponential random variable X with a uniform random variable as its parameter

$$X\ \sim Exp(U) ~ and\ U\ \sim U(0,1) $$ The question asked for the value of $ P(X\geqslant 1)$ I saw the solution and it went like this: $$P(X\geqslant 1) = E[P(X\geqslant 1)|U] = E[e^{-u}] = \int_{...
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How to change some parameters of a process to make it WSS?

Consider the R.P., $X(t)=Au(t-T)$ where, $$A \sim N(\mu,\sigma^2)$$ $$T \sim Exp(\lambda)$$ and $u(t)$ is the unit step function. If $A$ and $T$ are independent we'll have, $$E(X(t))=\mu(1-\exp\{-\...
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Some help on this question about stochastic epidemic models without removals would be greatly appreciated

I get this question up until the underlined part, would someone be able to explain the rest of it to me as its had me stumped for a little while now.
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Visualizing relationships in log-link/exponential distribution models by placing the linear predictor on the Y axis?

I'm visualizing results from a negative binomial regression. I don't want to the graph of Y vs X to look exponential, I want it to look linear. In SPSS, the value provided for the linear predictor is ...
2 votes
1 answer
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Light Bulb hypothesis testing

One claims that the life time distribution of its Everyday light bulbs is exponential with mean 1000 hours. If you test a random sample of 4 light bulbs and find that the average life time is 900 ...
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Find UMVUE of difference of parameters of two exponential distribution random variables

Let $X_{1}, \dots, X_{n}$ be i.i.d. having the exponential distribution $Exp\left(0, \theta_{x}\right)$ with $\theta_{x}>0$, and $Y_{1}, \dots, Y_{n}$ be i.i.d. having the exponential distribution $...
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Sum of exponential decays with normally distributed lifetimes

If I have the sum $$f(t)=\frac{1}{N}\sum_{i=1}^N e^{-\frac{t}{\tau_i}}$$ where each $\tau_i$ is distributed normally with mean $\tau$, what will be the mean functional form of $f(t)$ in the large $N$ ...
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What is $E[X]$ and $\text{Var}(X)$ if $X$ follows $Pois(T^2)$ and $T$ follows Exponential distribution [duplicate]

I'm new to this community. I have problem in finding expected value and variance of R.V.s that are composed of other R.V.s following other distributions. Suppose $X \sim Pois(T^2)$ where $T \sim Exp(\...
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1 answer
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Why my cdf of the convolution of n exponential distribution is not in the range(0,1)?

I assume that there are n exponential distribution that $x_i$ ~ $Exp(\lambda_i)$, i=1..n, and I want to calculate the cumulative distribution of $S=x_1+x_2+...+x_n$, the convolution of n exponential ...
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Equally spaced points on x-axis in exponential distribution probplot/qqplot

I have a sample distribution and want to check if it is exponential. For that I am trying a qqplot and probplot. ...
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Help understanding gamma posterior of exponential likelihood

The posterior of $\text{Exp}(x;\lambda)$ with prior $\text{Gamma}(\lambda;\alpha, \beta)$ is $\text{Gamma}(\lambda|\alpha+n, \beta + n\bar x)$ where $n$ is the number of observations and $\bar x$ is ...
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Expected value of maximum of $n$ iid exponential random variables [duplicate]

I was recently playing around with this distribution. Let $Y_n \sim \max_i X_i$ where $X_i \sim \exp(\lambda)$. Then the well-known result $$ f_{Y_n}(y) = \lambda n e^{-\lambda y}(1-e^{-\lambda y})^{n-...
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Distribution of a fraction of exponential random variables

Let $X_1, X_2$ be independent exponential random variables with common pdf $f(x)=\lambda\exp(-\lambda x), x>0$. How do I show that $Z=X_1/(X_1 + X_2) \sim U(0,1)$? I know that $F_Z(z) = P(\frac{X_1}...
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Exponential Posteriori with a Uniform Prior

I'm studyng for a final exam and found this problem from another generation, but I don't know how I should continue... I will be gratefull for any help, thanks you. Let be $X|\theta\sim U(0,\theta)$ ...
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How to use the exponential distribution to generate samples from the chi-square and beta distributions?

I am supposed to use rexp() in R to draw from an exponential distribution with mean 1, and then use those draws to generate 1000 draws from each of the following: ...
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Confidence interval for exponentially distributed estimator

We have an estimator $\hat{\theta}\geq 0$ for $\theta$, with distribution function $P\{\hat{\theta}\leq t \}=1-e^{-t/\theta}$, which we can recognize as the cdf of the exponential distribution. Our ...
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How to find an unbiased estimator for reciprocal of scale parameter given an iid exponential sample?

For a random sample $X_1, ..., X_n$ from an exponential distribution with scale parameter $\lambda$, the density is given by $f(x) = \frac{1}{\lambda}e^{-\frac{1}{\lambda}x}; \,x \geq 0,\, \lambda >...
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How to prove that t = min{t1, t2} follows an exponential distribution if t1, t2 follow another different exponential distributions

I have no idea about how to prove the next: Suppose we have two random variables, $t_1$ and $t_2$, that follow the distributions $\lambda_1e^{-\lambda_1 t_1}$ and $\lambda_2e^{-\lambda_2t_2}$, ...

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