Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [exponential-distribution]

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

6
votes
0answers
134 views

Poisson Process: indepedent increment

Let $\{N(t): t\geq0\}$ be a Poisson process of rate $\lambda$, and let $S_n$ denote the time until the $n_{th}$ event occurs. compute $P(S_3>5|N(2)=1)$ Attempt: Notice that $P(S_3>5)=P(N(5)&...
6
votes
0answers
142 views

Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
5
votes
0answers
115 views

First moments of Geometric Brownian Motion-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^...
5
votes
0answers
60 views

Find probability density function for $\varepsilon \cdot X$.

Let $ X $ be exponentially distributed with parameter $\lambda > 0$ . $a)$ Find the probability density function for $ Y := exp(X) $. $b)$ Consider $X$ again. Now let $\varepsilon$ be an ...
3
votes
0answers
1k views

Method of moments exponential distribution

Find the method of moments estimate for $\lambda$ if a random sample of size n is taken from the exponential pdf, $$f_Y(y_i;\lambda)= \lambda e^{-\lambda y} \;, \quad y \ge 0$$ $$E[Y] = \int_{0}^...
3
votes
0answers
1k views

Probability - long-run proportion

So the problem asks: A zoo owns a population of rare Galapagos turtles. The turtles bring in a lot of visitors but unfortunately do not breed in captivity. To maintain the population, new turtles ...
2
votes
0answers
49 views

How do we obtain an estimate for $\textbf{P}(X\geq 1)$ where $X\sim\text{Exp}(\lambda)$?

Consider a simple sample $X_{1},X_{2},\ldots,X_{n}$ whose distribution is given by $X\sim Exp(\lambda)$. (a) Determine an estimator for $\lambda$ according to the method of moments. (b) Determine ...
2
votes
0answers
82 views

Show that $P(N \geq n) = (1-e^{-\lambda})^n/ \lambda^n $

The lifetime $X$ (in days) of a device has an exponential distribution with parameter $\lambda.$ Moreover, the fraction of time which the device is used each day has a uniform distribution over the ...
2
votes
0answers
64 views

Average response/waiting time for aggregated tasks with Poisson arrival

Suppose there is a specific computation task with Poisson arrival rate $\lambda$ that could be aggregated in a way that when a task arrives and triggers a computation which lasts for $D$ seconds, if ...
2
votes
0answers
32 views

modelling arrival events for systems with multiple users?

Assume that the system at hand has a set of users M = {M1, M2, ...., Mn} and each Mi has a probability of creating an event e.g., M1=0.3, M2= 0.1, where the total probability for all users =1. The ...
2
votes
0answers
66 views

Distribution of exponential $x_i$s : $\sum_{i=1}^n x_i$

Exercise : Let $X_1, \dots, X_n$ be a random sample from the Exponential Distribution with unknown parameter $\theta$. (i) Find a sufficient and complete statistics function $T$, for $\theta$. ...
2
votes
0answers
55 views

Probability of two randomly selected leaves of a tree to be connected only at the root

Consider a bifurcating tree with $n>1$ leaves (or "individuals") produced by a branching process. All individuals derive from a single founder individual at the root of the tree, such that we can ...
2
votes
0answers
22 views

Why is it true that $t_i$ ~ exponential($\theta$) implies $\theta t_i$~exponential($1$)$=\frac12 \chi_2^2$

Why is it true that $t_i$ ~ exponential($\theta$) means $\theta t_i$~exponential($1$). Do we treat $\theta$ as a known constant, but why? Also, we know exponential($1$)=gamma($1,1$) and $\chi_2^2$= ...
2
votes
0answers
103 views

Exponential family distribution in terms of natural logarithm

I am studying a note on basic probability which introduced the exponential family as $\left\{p(x;\theta): \theta \in \Theta\right\}$ where $p(x;\theta) =h(x)c(\theta) \exp\left\{ \sum_{i=1}^{k}w_{...
2
votes
0answers
335 views

Expected time for the queue to become empty.

Question: Suppose, there is a queue A having i customers initially. The service time of the queue is Exponentially distributed with parameter $\mu$ (i.e., mean service time of one customer in Queue A ...
2
votes
0answers
33 views

Find the distribution of $R=\bar X / \bar Y$ where $X$ and $Y$ are exponentially distributed

Find the distribution of $R=\bar X / \bar Y$ given that exponential distribution with mean parameter 2 is equivalent to Chi squared with 2 dof. I'm given 2 samples $x_i$ and $y_i$ of sizes $n$ ...
2
votes
0answers
45 views

Showing random variables are independent

The question I am attempting is the one shown below: It is easy enough to find $F_U , F_V $ from definitions, and I have shown that $U \sim \epsilon (2) $ by subbing in to the expression for $F_U$. ...
2
votes
0answers
403 views

Poisson bus arrival process with different arrival rate

I want to know what I am thinking is right. There are two types of buses, B1 and B2. The arrival rate of B1 is r1 and B2 is r2. The arrival of those buses are independent Poisson arrival process. ...
2
votes
0answers
136 views

Find $ \mathbb P \{ \max(X, Y) - \min(X, Y) \gt 0.2 \} $.

Let $X$ and $Y$ be iid random variables for which $X \sim \text{Expo}(2)$ and $Y \sim \text{Expo}(3)$. Find $ \mathbb P \{ \max(X, Y) - \min(X, Y) \gt 0.2 \} $. SOLUTION: There is a clever way to ...
2
votes
0answers
39 views

An exponential family problem

I don't know how to express the problem to the form $f(x|\theta)=h(x)c(\theta)(\sum_{i=1}^{k}\omega_i(\theta)t_i(x))$ Let $X$ have pdf $f(x)=\frac{1}{\beta}e^{-(x-\alpha)/\beta}$, $x>\alpha$ ...
1
vote
0answers
12 views

Sufficiency in the exponential distribution

I am trying to show that given a random sample $\{X_i\}_{i=1}^n$ where $X_i\sim exp(\lambda^{-1})$, the statistic $T(\mathbf{X})=\sum_{i=1}^n X_i$ is sufficient by using only the definition. I have ...
1
vote
0answers
22 views

Exponential Test

Given a collection of data points, without any other information I want to test for an exponential distribution. In a different case, I tested for normality using an Accord implementation of the ...
1
vote
0answers
33 views

Probability $P\{\min{X_1,X_2}\leq X_3\}$?

Let $X_1, X_2$ and $X_3$ exponential random variables with the same parameter $\beta$? The PDF and CDF of $X_i$ are $$ f_{X_i}(x)=\beta e^{-x \beta}, $$ $$ F_{X_i}(x)=1-e^{-x \beta}. $$ The PDF and ...
1
vote
0answers
36 views

How do I find $E\lfloor X \rfloor$ when $X \sim \text{exp}(2)$?

I am trying to learn from my mistakes, and faced the following problem: Let $X \sim \exp(2)$ and $Y=\lfloor X \rfloor$, compute $E[Y]$. Well my false attempt was: First compute the PDF of Y: $ ...
1
vote
0answers
13 views

Density Function of Series Connection

The random lifetime of a certain electronic component is given by- $ f_Y(y) = \dfrac{1}{a} e^{-(y/a)}, y>0 $. Two such components are connected in series. The system fails when the first component ...
1
vote
0answers
32 views

Calculate the confidence interval of parameter of exponential distribution with summarySE in R?

I am trying to calculate the confidence interval for a set of data with the assumption they follow Exp dist. To achieve this, I am merging this with this in R, but does not work as I am not very ...
1
vote
0answers
30 views

Departure rate of Poisson tasks with aggregation

There is a single class of tasks with Poisson arrival rate λ at a processing node which takes a fixed-length time interval of D seconds to process a task. These tasks could be aggregated in a way that ...
1
vote
0answers
41 views

Two ice machine problem by using hypo-exponential distribution?

Ice machine 1 is currently working. Ice machine 2 will be put in use at a time $t$ from now. If the lifetime of ice machine $i$ is exponential with rate $\lambda_i$, $i = 1, 2$, by using hypo-...
1
vote
0answers
47 views

Distribution of Square root of Rayleigh distribution

I know that if $X\sim Exp(\lambda)$, then $\sqrt{X} \sim Rayleigh(1/\sqrt{2\lambda})$. Now I have $h^2 \sim Exp(1)$, then what is the distribution of $Z = [h^2]^{1/4}$. As the first step, I know ...
1
vote
0answers
20 views

Help with finding correct bounds of integral in exponential distribution.

The average time that a light bulb burns before it fails is 1000 hours. The probability that this light bulb burns between 100 and 1000 hours before failure is:____. (3 decimal places) $$λ = \frac{1 \...
1
vote
0answers
35 views

Wald's test. Hypothesis testing. Verification.

Let $X_1, . . . , X_n ∼ Exp(λ)$, for some unknown parameter $λ > 0$ and let $λ_0$ be a (known) fixed positive number. Consider the following hypotheses: $H_0 : ”λ = λ_0” vs. H_1 : ”λ ≠ ...
1
vote
0answers
17 views

M/M/1 queueing theory system

Could someone verify if my calculated parameters are right? λ = arrival rate = 0.8 Ws = average time spent in the system = 18/4 = 4.5 Ls = average number of customers in the system = 18/10 = 1.8 ...
1
vote
0answers
39 views

A problem in probability of exponential variables

I have a problem in solving a probability problem which does not seem to be hard to me but I can not get the correct answer. Let $X,\tilde{X}\sim \operatorname{Exp}(\lambda)$, $Y\sim \operatorname{...
1
vote
0answers
249 views

Prove standard cauchy distribution isn't part of exponential family.

So the density is: $$f(x) = \frac{1}{\pi(1+x^2)}$$ and I am supposed to prove this standard Cauchy distribution isn't part of the exponential family. I assume the best way to go about this would be ...
1
vote
0answers
33 views

Correctness of exercise and solution, related to a exponential distribution

I find this exercise somewhere The distance between two big fissures on a highway have an exponential distribution with mean $5$ miles. What is the probability that in a section of the road of $10$ ...
1
vote
0answers
231 views

Expected value (waiting time) of two independent exponentially distributed arrival times occuring in a given distance

Suppose we have two independent exponentially distributed arrival times X1, X2 having rates λ and μ. This means their corresponding expected waiting times are 1/λ and 1/μ accordingly. Now I'm ...
1
vote
0answers
74 views

Uniform distribution that depends on exponential?

So I'm trying to solve this problem: Let $X \sim \operatorname{Exponential}(\lambda)$. Let $Y \sim \operatorname{Uniform}(0,X)$. You are given $\operatorname{E}[X] = 1/ \lambda$ and $\operatorname{E}[...
1
vote
0answers
54 views

Joint density of partial sum of spacings and $Y_1$

Given $\ A_{(1)}, \ A_{(2)}, \ A_{(3)}, \ A_{(4)}, A_{(5)}\ $ are the order statistics of i.i.d random variables $A_i$ where $f(a_i|\alpha) = \beta^{-1}\ exp(\beta^{-1} (a_i-\alpha))$. Let $Y_{i} = (n-...
1
vote
0answers
30 views

Convergence in probability to some value

Find $a$ such that $X_n \xrightarrow[n\rightarrow \infty]{ P } a $, where $X_n = n^{-1} \sum^n_{i=1} Y_i $, where the $Y_i \sim Exp(2)$ are independent random variables. I tried: \begin{equation*}\...
1
vote
0answers
53 views

Show that a certain likelihood ratio test (for common mean of exponential RVs) is distribution free

For a dataset $X_1, ..., X_n$, the random variables are independent and $X_i \sim \text{EXP}(\theta_i)$. Consider the test where $H_0$ claims $\theta_1 = \theta_2 = ... = \theta_n = \theta$, while $...
1
vote
0answers
78 views

Solving Sum of Binomial Coefficients and Reducing to Exponential

I want to solve the following equation: $$\sum_{l=0}^{K-1} \binom{N-1}{l} {(1-\epsilon)}^l {(\epsilon)}^{N-1-l} \le \eta/\epsilon$$ and want to have expression of N as a function of $\epsilon, \eta$ ...
1
vote
0answers
496 views

Sum of exponential and normal random variables, X+Y, where the variance of Y is proportional to X

I have a question about an interesting convolution. Suppose the revenues produced by new product development projects are being forecasted. The revenue produced by a new product development project ...
1
vote
0answers
27 views

How to prove $\sum\limits_{k=1}^\infty \frac{k^N}{2^k}\leq c\mathbb{E}[X^N]$ where $X$ is exponentially distributed with parameter $\ln(2)$?

I came across the series $$A(N):=\sum_{k=1}^\infty \frac{k^N}{2^k},~~N\in\mathbb{N},N\geq 4,$$ which is the polylogarithm of order $-N$ evaluated at $\frac{1}{2}$. I got the hint that this can be ...
1
vote
0answers
186 views

Sum of infinite number of exponential random variables

Let $S_n \sim Exp(n)$ for $n = 1,2,...$ and define $\zeta = \sum_{i=1}^\infty S_n$. Is it true that $\mathbb{P}(\zeta = \infty) = 1$ ? We have $\mathbb{E}(\zeta) = \infty$ by a quick ...
1
vote
0answers
424 views

How the time between events of a sample of Poisson process is exponential but the time of occurrence is uniform?

I know that Poisson process is the probability of occurring an event in an interval, on the other hand, I know that the probability of the time of occurrences is uniform and the probability of waiting ...
1
vote
0answers
101 views

Exponential and poisson distribution

The manager of a video store noted the amount of time needed for customers to be served by the cashier. He concluded that the checkout times are exponentially distributed with a mean of 4 minutes. ...
1
vote
0answers
82 views

Independence and Mutual Exclusion of a Continuous Random Variable

I've come across an interesting question while studying for my final exams and was looking for some clarification. The Question Let X have a uniform Expo(3) distribution and let A be the event that ...
1
vote
0answers
105 views

Exponentially distributed events in time-partitions

The exponential distribution $p(t) = \lambda e^{-\lambda t}$ is a probability distribution that describes the time between events in a Poisson process. If $X$ is an exponential random variable, then ...
1
vote
0answers
116 views

PDF of $U = \frac{X}{X + Y}$

Just heads up. This is a homework question, so feel free to prod me towards the answer rather than answering it if you want :) With two random variables $X~\tilde{}~Exp(1)$ and $Y~\tilde{}~Exp(1)$, I ...
1
vote
0answers
59 views

Writing the complementary cumulative distribution as an expectation over indicator functions

Here is a neat little relation and I am wondering if/how it generalizes. The complementary cumulative distribution functions of a random variable, $X>0$, with density, $\rho(x)$, is $$C(x)=\int_x^\...