# Questions tagged [exponential-distribution]

To be used for questions on using, finding, or otherwise relating to Exponential Distributions.

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### Excess waiting time given two exponential variables

Suppose that there are two patients that arrive on time to a clinic. The time that each patient takes with the doctor is distributed according to an exponential and the expectation of that is 0.5 ...
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### Solving a stochastic equation by characteristic functions

Based on the work of Nicolas Curien and Takis Konstantopoulos titled Iterating Brownian motions, ad libitum I would like to prove the following (based on the last paragraph of the proof of Proposition ...
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### Independence of maximums of independent random iid exponential variables.

Given Random Variables $X_1,...,X_n \sim Exp(\lambda)$, I want to show that for $m \neq n$ the sets $\{X_n = \max_{i=1,...,n} X_i \}$ and $\{X_m = \max_{i=1,...,m} X_i \}$ are independent. My first ...
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### Simulate a Brownian motion by exponential time stepping

Let $(B_t)_{t\ge0}$ be a Brownian motion. We can simulate a path of $(B_t)_{t\ge0}$ using the Euler-Maruyama discretization scheme. Now, in the paper Efficient Numerical Solution of Stochastic ...
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### How to find the power function given exponential distribution?

Let $X$ be distributed according to $Exp(θ)$ and $H_0: θ = 1$ and $H_1: θ = 5$. We have a test that rejects the null hypothesis if $X < 0.05$. Determine the power of the test. In the answers they ...
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### Turning Nearest Neighbour Distribution of Poisson Scatter Theorem to Rayleigh Distribution By Multiplying Constant

Consider a Poisson random scatter of points in a plane with mean intensity $\phi$ per unit area. Let R be the distance from 0 to closest point of the scatter. Show that $\sqrt{2\phi\pi}$R has the ...
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### How do I apply the Rao-Blackwell Theorem to find MVUE of parameter theta?

Let Y1, Y2, . . . , Yn be independent and identically distributed random variables having the same population distribution with density: f(y; θ) = ( θ(3^θ)/y^(θ+1) , y ⩾ 3; 0, elsewhere.) where θ is a ...
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We know that that given $X_1,...,X_n$ independent random variables which are distributed exponentially, their minimum is also distributed exponentially. Is the converse true? Given $X = \min\{X_1, ...,... • 135 0 votes 1 answer 34 views ### Memoryless property with any random wait time We see here a proof of the random-time memoryless property$P(X>T+s|X>T)=P(X>s)$where$E\sim Exp(\lambda)$and$T\ge 0$is a continuous random variable independent of$E$. The proof, however,... • 197 0 votes 0 answers 23 views ### Derivation of Gamma distribution without using Poisson distribution Most of the derivations of the Gamma distribution pdf I've seen on here use the Poisson distribution. My lecture notes use the Gamma distribution and the exponential inter-arrival time definition of a ... • 197 0 votes 0 answers 12 views ### Understanding discrete vs continuous density rates Imagine I have an array of sites ($1\leq i \leq N$) that "activate" at given activating rates$\{a_i\}$, so that the time$t_i$it takes for a site$i$to activate follows an exponential ... • 3,435 1 vote 1 answer 104 views ### Deriving the PDF of a function of two random variables Let$X$and$T_1, \ldots, T_m$be independent random variables following exponential distributions with parameters$\lambda_X, \lambda_T$, respectively. Let $$T_{\rho} = T_1 + \ldots + T_m$$ so that ... • 3,468 0 votes 1 answer 41 views ### Distribution of the sum of$m$inverse exponential random variables Let$T_1, \ldots, T_m$be$m$random variables with$T_i \sim \exp(\lambda_T)$. We are interested in the sum $$T' = \sum_{i=1}^{m}\frac{1}{T_i}$$ which is itself a random variable. By definition, ... • 3,468 0 votes 1 answer 132 views ### Finding the joint density function of$(X+Y,X)$with$X$and$Y$independent and following an exponential distribution with parameter$\lambda>0$Given two independent random variables$X$and$Y$that both follow an exponential distribution of parameter$\lambda > 0$, I am trying to find the joint density function of$(X+Y,X)$. I have ... 0 votes 0 answers 19 views ### How to use cumulative distribution functions within an interactive simulator I am building a simple simulator in python that should simulate an event taking place based on its ... • 101 0 votes 0 answers 67 views ### Suppose the length of life of certain kind of light bulb, after it is installed, is exponentially distributed with a mean length of 7 days Suppose the length of life of certain kind of light bulb, after it is installed, is exponentially distributed with a mean length of 7 days. As soon as one bulb burns out, a similar one is installed in ... 2 votes 5 answers 95 views ### Does anyone have any general rules of thumb for when to use$e^x$vs$2^x$(or any other non-special exponential) during modeling? Recently, I picked up a book about statistics and probability. I know$e^x$is special because of its derivative and corresponding growth rate. However, I have a hard time connecting this academic ... • 4,085 1 vote 0 answers 19 views ### Type I Error Rate Higher than Significance Level in Likelihood Ratio Test for Exponential Distribution Type I Error Rate Higher than Significance Level in Likelihood Ratio Test for Exponential Distribution Problem Description I am conducting a Likelihood Ratio Test (LRT) to determine if data from a two-... 2 votes 0 answers 42 views ### Simplifying equation with summation and integrals Let$X$be a uniformly distributed random variable between 0 and 10. Also let$Y$be exponentially distributed with$\lambda=1$. I want to solve the following equation where$\tau\geq 0$, $$\sum_{i=1}^... • 427 3 votes 1 answer 54 views ### Statistics Question of the Day Motivation: I am a graduate student in the Department of Statistics at Kansas State University. Everyday I create a "question of the day" for myself, and it has been going well for the past ... 0 votes 0 answers 80 views ### A natural exponential family Consider a discrete probability distribution with the following probability mass function$$f(y;\lambda ,ν) =\frac{\lambda^y}{A(\lambda,ν)(y!)^ν}$$,$y= 0,1,2,...,$with parameters$\lambda \gt 0,ν \...
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I have encountered a problem with the variance of Poisson and exponential distribution. Suppose there is an accident that follows a Poisson distribution with an occurrence rate of $\lambda$ per hour. ...