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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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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1answer
65 views

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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1answer
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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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1answer
21 views

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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1answer
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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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35 views

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 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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19 views

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
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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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1answer
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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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Significance level of a continuous distribution

Can the significance level of a continuous distribution be exactly equal to its value? For example for Exponential Distribution can the significance level of a test be exactly 0.02?
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Non-linear regression of exponentially distributed data?

I have a family of random variables $y(x)$ and a set of data points $(x_i, y_i)$. I know that, for each $x$, $y(x)$ is exponentially distributed. I have a hypothesis that the mean of the distribution ...
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46 views

Minimizing Mean Square Error

Suppose we have a random sample $\textbf{X}=(X_1,...,X_n)$ from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & ...
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Flat “geometric distribution” by varying the probability of the Bernoulli trail

In a simulation I am working on, each day (time step) there is a chance that a condition changes (at which point it is stuck in the changed condition). Setting this probability to a fixed value (say 5%...
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generating a uniform random variable from the first digit of an exponential random variable?

in "introduction to probability models", Ross talks about simulating with the rejection method, and he needs an exponential random variable, and a uniform random variable (used only for checking ...
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Conditional and joint distribution of the sum of exponential RVs

Let $X_1,X_2,...,X_n$ be i.i.d. $Exp(\lambda)$ random variables and $Y_k =\sum^{k}_{i=1}X_i$, $k = 1,2,...,n$. a) Find the joint PDF of $Y_1,...,Y_n$. b) Find the conditional PDF of $Y_k$ ...
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Sensitivity analysis with an exponential distribution

Statistics newbie here: If you had to run a local sensitivity analysis for an exponential distribution, would you rather change the λ (e.g. with a 5% increment) or would you take your initial value (...
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1answer
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Finding the pdf of $X_{(1)}$ of the two-parameter exponential distribution

I have to find the pdf of the smallest order statistic $X_{(1)}$ of two-parameter exponential distribution whose pdf is: $f(x; \theta_1, \theta_2) = \frac{1}{\theta_2} \exp\{-\frac{x-\theta_1}{\...
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1answer
59 views

Transformation of a random variable with a gamma distribution

Suppose $X_i \stackrel{i.i.d}{\sim}$ Exp$(1/\theta)$ which implies $\sum_{i =1}^{n} X_i \sim$ Gamma $(n, 1/\theta)$. But, then, the book that I am reading says that $(2/\theta)\sum_{i =1}^{n} X_i \...
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Local sensitivity analysis with exponential and uniform distributions as input

For my thesis I need to run a sensitivity analysis on the input factors for a supply chain model. I am supposed to change the mean and the standard deviation (sd) of all input factors respectively by ...
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1answer
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Estimating mean values given a histogram (without using median point)

I have a list of stock trades made by US senators and the exact amount of money they spend on a trade is not disclosed, they only give a range of values. Below is the frequency table for the 8600 ...
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1answer
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Variance of $\frac{X_i}{\theta^2} -\frac{1}{\theta}$ in an exponential distribution

I read in a book discussing the exponential distribution that the variance of $\frac{X_i}{\theta^2} -\frac{1}{\theta}$ is equal to $\frac{1}{\theta^4}Var(X_i) = \frac{1}{\theta^4}\theta^2$. Can ...
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2answers
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What is the distribution of a mixture of exponential distributions whose rate parameters follow a gamma distribution?

I want to know the theoretical distribution of a mixture of exponential distributions whose rate parameters are distributed according to a gamma distribution: $$ y\sim\text{Exp}(\theta), \quad\text{...
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Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...
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Maximums of two exponentials

The wikipedia page for the exponential distribution states that for $X_1, X_2, \dots X_n$ independent exponentially distributed with rate parameters $\lambda_1,\lambda_2,\dots,\lambda_n$, the index of ...
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1answer
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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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1answer
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Sample Size for Exponential Distribution

I'm having a bit of trouble with a simple question. I have data for a product, amount and days on which it was sold, that when plotted as a cumulative histogram follows an exponential cdf. I want to ...
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1answer
314 views

Sum of Exponential and Gamma Distributions [duplicate]

I have been learning sums of distributions and understand that the sum of exponential distributions with parameter B is a gamma distribution with parameters a=1 and B. However, I need to figure out: ...
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2answers
273 views

How could one (tentatively) estimate the actual number of COVID-19 infections in an area, using hard data like age-adjusted death rate?

A big problem with the current COVID-19 epidemic is the difficulty of getting tested (due to mild symptoms and lack of testing kits). This makes it impossible for patients with little or no symptoms ...
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1answer
74 views

Linear Model, Distribution of Maximum Likelihood Estimator

Let $\epsilon_i \sim \text{Exp}(\lambda)$, $\lambda > 0$, and iid for all $i = 1,2, \dots$ Suppose for we have the linear model $$ Y_i = \beta X_i + \epsilon_i,$$ where $X_i > 0$ for all $i =...
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Distribution of the median of an exponential distribution?

Say i have $X_{ij} \sim Exp(\theta)$ for $i=1...n,\;\;j=1...k$. Afterwards I derive $k$ medians for $k$ groups of size $n$ (even number). The median is a random variable say $M_j$. Do we have any ...
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Quantile-calculation example: Why the term $c$?

I need help to interpret a solution to the following example: Based on historical claim amounts $x_1,...,x_{47}$, which are assumed to be outcomes from an unknown claim distribution, you are ...
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3answers
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Fitting data to an exponential model with a specified rate in R

I have been using the fitdistr package in R to try and do this but with no luck so far: fit1 <- fitdistr(data1$x, "exponential", start = list(rate = 10)) I am ...