Questions tagged [exponential-distribution]

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

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Let $X$ be an exponential random variable and $\Bbb{P}(X \in A) < \Bbb{P}(X \in B)$. Is $\Bbb{P}(aX \in A) < \Bbb{P}(aX \in B)$ for $a > 0$?

Let $X$ be an exponential random variable (say with mean 1) and $\Bbb{P}(X \in A) < \Bbb{P}(X \in B)$ for two events $A, B \subset\Bbb{R}$. Is $\Bbb{P}(aX \in A) < \Bbb{P}(aX \in B)$ for $a > ...
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Why does the unbiased statistic in this example be MVUE immediately?

I am reading "Introduction to Mathematical Statistics" edition 8 by Robert V. Hogg et al. to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. ...
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Conditional distribution of one of the two exponential random variables, given one is smaller than the other

Let $X$ be a random variable with exponential distribution with parameter $a$, i.e. $X\sim Exp(a)$. See https://en.wikipedia.org/wiki/Exponential_distribution for the definition of exponential ...
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What is the probability that the time until the machine requires service exceeds 60 days [closed]

What is the probability that the time required exceeds 60days? Suppose the time in days between service calls on a photocopier machine follows an exponential distribution with a mean call of 0.02
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Deriving the geometric distribution from memorylessness

It is "well-known" that there are basically only two types of memoryless distributions: The exponential distribution (in the continuous case) and the geometric distribution (in the discrete ...
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Expectation of Exponential Random Variable

I was a bit confused about this question. 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 ...
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Calculating Expected Number from Exponential Distribution

I was a bit confused about this question. 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 ...
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Sequence of an exponential random variable

Given that $X_n$ is an exponential random variable with parameter $λ=n$. How does $P(X_n≥ε)=e^{-nε}$ ? According to this probability course, the equality holds since $X_n∼Exponential(n)$. Honestly, ...
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Finding optimal (=lowest cost) roads connecting all cities where cost of roads are iid exponentially distributed (minimum spanning tree)

I’m trying to solve/understand some exercises about problems involving exponential distributions. The professor in the following video (https://www.youtube.com/watch?v=DfROPYAjbfM&list=...
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Conditional distribution of $X$ given that $X+Y > t$. [closed]

If $X$ and $Y$ are independent exponential random variables with the same mean, then what is the conditional distribution of $X$ given that $X+Y > t$? Form comment: I was able to get that the ...
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Continuous-time Markov chains: system with components in parallel and enough repairmen. How to calculate availability?

I am trying to solve a continuous-time Markov chain exercise. But I have been stuck for a few days and I don't know how to continue or even If what I have done is correct. Consider a system having ...
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Gibbs sampling - finding probabilities of joint exponential random variables

Say I have two IID exponential random variables, both with mean $\lambda$. I would like to find p(x < a | x+y > b) for two positive integers a and b using Gibbs sampling. I understand that for ...
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Alternating Renewal Process: How to calculate variance without knowing how the two distributions depend on each other

I am trying to solve a Alternating Renewal Process exercise. The "on" state follows a exponential distribution with mean 2. The time in the "off" state follows a gamma distribution ...
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why do I have to use normal distribution in this exercise?

on average, a biker inflates their bike tires every 8 days. the interval between each time, namely t1, t2, and so on, is an exponential random variable. Find probability that 40 inflations are ...
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(How) Can the typical continuous probability distributions be derived from elementary distributions?

There are certain probability spaces which I regard as very basic and very easy to match with physical intuition. These are for example: The Bernoulli $\mathrm{Ber}_p$ distribution with paramter $p$ ...
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Expected time until k failures among n items each independent and having exponential distribution

This is the question I am trying to solve. "Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution ...
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infinitesimal generator of a Markov process

Problem: There are $n$ identical components in a system that operate independently. When a component fails, it undergoes repair, and after repair is placed back into the system. Assume that for a ...
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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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Evaluating the integral $\int e^{\kappa \mu^Tx}dx$ when $dx = \prod_{j=2}^{p-1}\sin^{p-j}(\theta_{j-1})$ in von Mises-Fisher distribution

While reading a dissertation Matrix nearness problems in data mining (link) I noticed that the author derives the normalization constant of the von Mises-Fisher distribution (link): $c_p(\kappa)e^{\...
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Knowing if two random variables are independent or not [closed]

Consider three independent random variables $X$, $Y$, and $Z$. Here, $X$ and $Y$ are i.i.d exponential random variables with parameter $\lambda$, and $Z$ is an exponential random variable with ...
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The Cauchy-Stieltjes transform of the exponential distribution

I'm interested in the integral $$G(z)=\int_\mathbb{R} \frac{ce^{-cx}\mathrm{d}x}{z-x}$$ which defines the Cauchy-Stieltjes transform of the exponential distribution parameterized by some real $c>0$....
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one-dimensional regular exponential family

I have got the question: Let $X$ and $Y$ be independent and exponentially distributed random variables with $E(X) = \mu$ and $E(Y) = 2\mu$, where $\mu > 0$. Represent the family of joint ...
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Finding lambda in exponential distribution

I have the time data I hold in hours for 15 days. I want to calculate P(x>1.5) with exponential distribution in line with these data, but I am not sure how to find lambda value. Should it be (the ...
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independence of exponential random variable and general independency

Maybe this is not a good enough question, but a long search on the internet led me to post the question here. Let's say we have three independent Poisson processes that are essentially the "...
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How to solve the probability that customer C1 is the last to withdraw from the bank?

Consider a bank with two clerks. Three people, $C_1$, $C_2$, and $C_3$, enter simultaneously. $C_1$ and $C_3$ go directly to the clerks, and $C_3$ waits until either $C_1$ or $C_2$ leaves before he ...
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Sum of exponential variables with decreasing means

Problem Let $Y_k$ be exponential indipendent random variables of parameter $\alpha k$ for $k\in \{1,2,...,n\}$, i.e. the CDF is $P(Y_k\le x)=1-e^{-\alpha k x}$ Prove that the CDF of $$S_n=\sum_{k=1}^...
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Question about the CLT as it applies to the exponential distribution

The question at hand is as follows... The time intervals (measured in hours) between arrivals of emails to your mailbox can be modelled as i.i.d. exponential random variables with parameter 5, namely $...
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Calculating the probability that two or more clocks ring

Given a set of iid random Variables $X_n\sim \exp(\lambda)$. We have a (countably) infinite number of clocks $c_1,c_2,...$. Let $t=T$ be the maximum time. When our time starts (at $t=0$) the first ...
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Find posterior distribution of Poisson process knowing that the prior is Exponential$(1)$.

This is the problem: Bus arrival times form a Poisson process with intensity 𝜆 measured in buses per hour. Your prior distribution on 𝜆 is that 𝜆 is an exponential random variable, Exponential$(1)$....
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Measuring half-lifetime from 1-cumulative or frequency distributions, what's the difference?

I have a question that is blowing my mind. Let's say that I measure a phenomenon that has a duration in seconds. I can graph the data as a frequency distribution (a histogram), showing a nice ...
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Probability question on service time exponentially distributed

A post office is run by two clerks. When Smith enters the system, he finds that Jones is being served by one of the clerks, and brown is being served by the other. Also suppose that Smith is told that ...
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Proving the expectation of a stochastic process

Problem Information packets arrive at a server with a poisson process having rate $\lambda = 2$ per hour. The server processing time for a packet follows the distribution : $f(x) = 1, 0\leq x\leq1$ ...
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How to find the CDF of $Y=\frac{1}{2}X$?

Let $X$ have an exponential distribution with rate parameter $\lambda=\frac{1}{2}$ I believe that the probability density of $X$ is $$f_X(x) = e^{-x/2}$$ So the CDF of $X$ is then $$F_X(x)=-2e^{-x/2}$$...
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Conditional MGF of the difference of two exponentially distributed random variables

let $X$ and $Y$ be two exponential random variables with the same parameter $\lambda$. I try to calculate the MGF of random variable $V=X-Y$ given that $X>Y$. We know that the MGF of Exponential ...
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Independence of product of random variables

Suppose I have three independent random variables $X, Y, Z$ that are all exponentially distributed with not necessarily different parameters $\lambda$. How can I show that $X\cdot Y$ and $Z$ are ...
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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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Mean of an Exponential Distribution whose rate parameter is also exponentially distributed

Suppose I have a random variable $X$ with an exponential distribution with rate parameter $\lambda$. Suppose also that I don’t know the value of $\lambda$ but that it will be drawn from another ...
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2 votes
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Deriving a mixed distribution from exponential and inverse gamma

Question You are given the following: the amount of an individual loss in the year $2022$ follows an exponential distribution with mean $15000$ Between $2022$ and $2025$, losses will be multiplied by ...
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Expectation of an exponentially distributed random variable

The question is as follows: Let $X$ be exponentially distributed with parameter $\lambda$, Find $\mathbb{E}[e^{sX}]$, where $s$ is a real parameter. For what values of $s$ does the expectation exist? ...
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1 answer
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Given random variable $X\sim \mathrm{Exp}(λ)$, what's the CDF of $Y=\cos(\pi X)$? [closed]

Given random variable $X\sim \mathrm{Exp}(\lambda)$, what's the CDF of $Y=\cos(\pi X)$ ? I know that the steps should be $F(Y)=P(Y\leq y)=P(\cos(\pi X)\leq Y)$, but the following steps are a bit ...
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Distribution of ordered independent exponential random variables

I'm working on the following exercise from Achim Klenke's "Probability Theory: A Comprehensive Course" (3rd Ed, Exercise 15.1.3): Let $n \in \mathbb N$ and let $X_1, \ldots, X_n$ be i.i.d. ...
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Distribution of first jump time

Let $(X_t)_{t\ge0}$ be a Lévy process, $\tau_0:=0$ and $$\tau_n:=\{t>\tau_{n-1}:\Delta X_t\in B\}\;\;\;\text{for }n\in\mathbb N$$ for some measurable set $B$ with $0\not\in B$. How can we show that ...
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exponential power distribution

I see two different definitions of the exponential power distribution: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf $$f(x)= \exp(1-\exp(\lambda x^k)) \exp(\lambda x^k) \lambda k ...
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What is the joint distribution of exponential random variables $T_1$ and $T_2$ when they are related as $T_2 = \max(T_1 - A_2, 0) + S_2$?

Suppose $T_{1}$ and $T_2$ are identically distributed RVs according to an exponential distribution of rate $\mu - \lambda$, $A_{2}$ is an exponential random variable with rate $\lambda$ and $S_2$ is ...
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If I have hypoexponential interarrival times, then does it fulfill a poisson process?

My doubt arose when I wanted to add two independent exponential interarrival times of parameter $\lambda$ and $\gamma$ respectively, with $\gamma \neq \lambda$. I know that the sum of two independent ...
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Finding correlation coefficient between transformed random variables.

If we have three iid random variables $X_1$,$X_2$,$X_3$ with common pdf f(x)= $e^{-x}$ for x greater than 0. How can we calculate the correlation coefficient between $Y_1$=$X_1$/$X_2$ and $Y_2$=$X_1+...
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How to solve this problem? (exponential distributions)

Studying for a re-take exam in a first term probability course and, among other things, I'd like to understand the solutions to the maths problems I didn't understand on my first attempt. I couldn't ...
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"Convexity" of a family of distributions

Let $\alpha\in(0,1)$. Consider a family of CDFs $\mathcal{X}$ that contains every CDF $X$ defined on $[0,\infty)$ with increasing hazard rate (IHR) which satisfies $$\mathbb{P}[x>y]=\alpha,$$ where ...
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Functioning Time vs Lifetime of Light bulb

Lifetime of a light bulb follows exponential distribution with $x$ means hours. If $n$ bulbs were switched on at same time and after $t$ time only $n-m$ were found to be functioning. Remaining $m$ ...
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Is this the correct way to use the Moment Generating Function? [duplicate]

Let Sk = X1+X2+· · ·+Xk such that Xi for i ∈ {1, 2, . . . , k} are independently and identically exponentially distributed random variables with rate λ, i.e. Xi ∼ Exp(λ). Prove with adequate reasoning ...
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