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Questions tagged [exponential-distribution]

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

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Consistency of a hypothesis test

I have given statistical model $((0,1)^n, \mathcal{B}(0,1)^n,\mathcal{P}_n)$, where $\mathcal{P}_n=\{ P_{\theta}^{\otimes n} \ |\ \theta \in (0, \infty) \}$ and each $P_{\theta}$ has density function $...
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Related to PDF of the sum and product of exponential random variable

In my research work, I came across the following situation that involves sum and product of exponential random variable as shown: $P = X_1 + \beta_0^2 X_2X_3$ ---(1) where $\beta_0^2$ is constant and $...
Heretolearn's user avatar
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Interpretation of a PDF

I have the probability density function (PDF) $$\frac{6}{5}(e^{-2x}+e^{-3x}),x>0 \text{ and 0 otherwise}$$ If I compare this to the two individual component PDFs: $$2e^{-2x},x>0 \text{ and 0 ...
Starlight's user avatar
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Laplace transform of probability density functions

I have a curious observation motivated by some applications of queueing theory. Let $f:[0,+\infty)\to\mathbb R^+$ be a PDF so that $\int_0^{+\infty}f(t)dt=1$. Let's say that $f$ is bounded and ...
Yining Wang's user avatar
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If $\text P[s+t<τ]=\text P[s<τ]\text P[t<τ]$ and $\mathcal F_t:=σ(\bigcup_{s\le t}\{τ\le s\})$, can we factorize $\text P[s+t<τ\mid\mathcal F_s]$?

Let $\tau$ be a $[0,\infty)$ valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$\operatorname P[s+t<\tau]=\operatorname P[s<\tau]\operatorname P[t<\tau]...
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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 ...
One's user avatar
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Can we simplify the integral $\int_0^\infty\cdots\int_0^\infty f\left(\sum_{i=1}^ns_i\right)g\left(\sum_{i=1}^n\alpha_is_i\right)ds_n\cdots ds_1$?

How can we simplify the integral $$I:=\int_0^\infty\cdots\int_0^\infty f\left(\sum_{i=1}^ns_i\right)g\left(\sum_{i=1}^n\alpha_is_i\right){\rm d}s_n\cdots{\rm d}s_1,$$ where $\alpha_1,\ldots,\alpha_n&...
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Joint Central Moments of Bivariate Exponential Distribution

While doing a study on the growth metrics RRRGR (Reverse of Relative of Relative Growth Curve), I'm stuck on the exact distribution of the growth metrics $ W_{ji} = ln \left( \dfrac{R_{ji} (t)}{R_{ji} ...
Pri's user avatar
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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 ...
user1047209's user avatar
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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 ...
undergradstudent123's user avatar
6 votes
1 answer
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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 ...
Need_MathHelp's user avatar
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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 ...
BurgerMan's user avatar
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Integral with two Exponential Distributions

X~exp($\lambda$) and Y~ exp($\mu$) are two independent variables. I am trying to figure out how to solve for $E[X^{\theta -1} 1_{X<Y}]$. I believe this could be expressed as: $$ \int_{0}^{\infty}\...
d123's user avatar
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Maximum Likelihood Estimation of median for an exponential distribution

Given data x1, ... xn i.i.d. with exponential distribution and unknown parameter λ, determine maximum likelihood estimation of θ given the observed data where theta is the median of the distribution. ...
Michael Williams's user avatar
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Function of exponential random variables

I have $20$ exponential random variables with mean $\alpha$ representing delays $D_n: n \in\{1,\ldots, 20\}$. I have 20 random variables denoting power $P_n^{'}: n \in\{1,\ldots, 20\}$. which depends ...
wanderer's user avatar
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Formal proof of joint pdf for arrival times of a Poisson process

Consider a probability space $(\Omega,\mathscr{F},\mathbb{P})$ which supports a Poisson process $N$. Let $T_1$ and $T_2$ be the first two arrival times from $N$, while $\xi_2$ is the first inter-...
Morris Fletcher's user avatar
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convolution of an order statistics and an exponential distribution

Is there a simplification to the convolution of the k-th order statistics from an erlang distribution with shape 2: $f_{1_{(k)}}(z) = \frac{n!}{(k-1)!(n-k)!} \cdot \left(1-e^{-\lambda z} (\lambda z+1)...
user9467051's user avatar
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Order Statistics from a sum of exponential distributions

Let $X_i$ $(X_1, \dots, X_n)$ and $Y_i (Y_1, \dots,Y_n)$ be i.i.d. exponential r.vs with rate $\lambda$. Let $Z_i= X_i+Y_i$. How to write the pdf of the k-th order statistics of the $Z_i$ random ...
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Prove a Result about Optimization and Exponential Distribution: Intuitive but Challenging

Let $v=(v_1,v_2,...v_N)$ be a random vector with $N$ elements. $v_i \perp v_j$ for any $i\neq j$. For each $i$, the CDF of $v_i$ is: $F_i(v_i)=1-e^{-(v_i-a_i)/\theta}$, where $v_i\in [a_i,\infty)$. ...
math101's user avatar
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Is $X = \max \left\{X_1, X_2\right\}$ exponential variable given $X_1$ and $X_2$ are?

I know that if $X_1$ and $X_2$ are independent exponential variables with parameters $\lambda_1$ and $\lambda_2$ then $X = \min \{X_1, X_2\}$ is exponential, and with $\lambda = \lambda_1+\lambda_2$. ...
Danny Wen's user avatar
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Exponential distribution and random sampling

Statement It's night and you are looking into the sky waiting to see a falling star. After the first hour you still haven't seen anything, so you check online and find two sources $s_1$ and $s_2$. ...
andreg's user avatar
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Particular case of Exponential Distribution Memorylessness

I have a dispute with my uni colleagues. There is an exercise requiring us to calculate, given an exponential distribution S, the following probability $$P(1<S<2|1<S).$$ According to them the ...
andreg's user avatar
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Rearranging exponentially distributed random variables by size

Let $X_1,\ldots,X_n$ be i.i.d. random variables, exponentially distributed with parameter 1. We rearrange the variables in a nondecreasing order, i.e. we find indices $i_1,\ldots, i_n$ such that $X_{...
Hernán Ibarra Mejia's user avatar
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Joint distribution of exponential random variable conditional on their sum

Let $Y_1, ..., Y_n$ be independently exponentially distributed with rate $\lambda$. My question: what is the joint distribution of $Y_1, ..., Y_n$ conditional on $T = \sum^n_{i=1}y_i$ ? We know that $...
Guillaume F.'s user avatar
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Show for $\tau (\theta) = \exp(-\theta c), c>0$ the estimator $E(X) = (\max(0, 1-\frac{c}{\sum x_i}))^{n-1}$ is efficiency unbiased. [closed]

TASK: Let $X = x_1, ... x_k$ are i.i.d. with exponential distribution with parametr $\theta$. Show for $\tau (\theta) = \exp(-\theta c), c>0$ the estimator $E(X) = (\max(0, 1-\frac{c}{\sum x_i}))^{...
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"log of a Pareto random variable has an exponential distribution"

This twitter exchange below: I'm partial to Taleb's (@Nero's) concepts, but was unfortunately not exposed to a lot of high-quality math in my childhood. As a result, I struggle to understand concepts ...
thanks_in_advance's user avatar
1 vote
1 answer
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Minimum record of exponentials getting broken finitely/infinitely many times

Let us consider the following scenario: we have a sequence $(X_n)_{n\geq 1}$ of independent random variables, where for every integer $n\geq 1$, $X_n$ is exponentially distributed with parameter $\...
sicmath's user avatar
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0 answers
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Integral involving exponential function and lower incomplete gamma function

Can we get the closed form value of the integral \begin{equation*} \int_{0}^{\infty}e^{-ax}\gamma(x,b)dx, \end{equation*} here, $a$ is a positive real numbers and $b$ is positive integer. Any ...
Naveen Kumar's user avatar
4 votes
1 answer
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Random walk where Increments have exponential distribution. Probability of never reaching a negative value after $n$ steps.

Consider the random walk $S_n = \sum_{i=1}^n (X_i-1)$ where $X_i$ are i.i.d. with exponential distribution and mean $1$, i.e. $P(X_i \leq x) = 1-e^{-x}$. I am trying to figure out the probability $p_n$...
Sebastian's user avatar
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does this expression look familiar (natural log with exponential)?

The certain payoff in a decision problem I am solving is affected by decision variables $k$ and $n$ as follows: $$f(k,n)= \frac{\alpha}{\lambda} \ln \left(\frac{\alpha + n e^{\frac{k}{\alpha}} }{\...
John Ritz's user avatar
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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 ...
VoidzenNullscape's user avatar
1 vote
2 answers
101 views

If minimum of independent random variable is exponential, is each one exponential?

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, ...,...
mathematico's user avatar
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1 answer
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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,...
hegash's user avatar
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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 ...
hegash's user avatar
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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 ...
sam wolfe's user avatar
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1 vote
1 answer
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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 ...
lafinur's user avatar
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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, ...
lafinur's user avatar
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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 ...
Gabriel Gontier's user avatar
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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 ...
FTM's user avatar
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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 ...
Gadin Naidu's user avatar
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 ...
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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-...
BINGXIN YAN's user avatar
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}^...
TK99's user avatar
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3 votes
1 answer
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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 ...
aidan kerns's user avatar
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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 \...
User1's user avatar
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1 vote
2 answers
109 views

Variance of Poisson and Exponential distribution [closed]

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. ...
Ziyao Zhang's user avatar
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Sum of Cumulative Max of Exponentially Distributed Variables

Let $X_1, X_2, ..., X_n$ be independent, identically and exponentially distributed random variables, $P(x) = k \exp(-k x)$. Define $Y$ as the sum of the sequence of cumulative maxima: $Y = X_1 + \max(...
GingerBreadMan's user avatar
4 votes
1 answer
146 views

How to apply queuing theory to find the long run proportion of customers who leave the system?

I am trying to apply queuing theory / birth and death process to the following. Suppose customers arrive in a restaurant according to a Poisson process with rate $\lambda = 1$. Suppose there are $2$ ...
MilesToGo's user avatar
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1 answer
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To what family of densities does $e^{-u}(u^k-k!) \log u$ belong?

Apparently $\int_0^{\infty} e^{-u} (u-1) \log u du = 1$, $\int_0^{\infty} e^{-u} \frac{1}{3}(u^2-2) \log u du = 1$ etc. Does these densities belong to a known family of densities? The closest I found ...
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