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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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Derive the probability density function of $Z = T_1 + T_2$.

Let $T_1$ be the waiting time until the first call in a call center and let $T_2$ be the waiting from the first call until the second call. Assume that both $T_1$ and $T_2$ have an exponential ...
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Searching for proof - bayesian inference for exponential distribution

According to Wikipedia (https://en.wikipedia.org/wiki/Conjugate_prior) the gamma distribution is a conjugate prior for the exponential distribution (with unknown rate-parameter, $\lambda$, and ...
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Find $E(\text{min}(X_1,X_2,X_3))$ where each $X_i$ is exponential with parameter $i$

I want to calculate $E(\text{min}(X_1,X_2,X_3))$ where $X_1\sim\text{exp}(1), \ X_2\sim\text{exp}(2)$ and $X_3\sim\text{exp}(3).$ Denote $M=\text{min}(X_1,X_2,X_3)$. By independence of the $X_i$ ...
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Uniform and exponential distribution

Consider an experiment. The duration of the experiment has uniform distribution on $[2,6]$h. When the experiment starts, the device A turns on. This device will turn off after time,which has ...
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1answer
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Minimum variance unbiased estimator of exponential distribution

The given model is $\text{Exp}(\mu,\sigma),\;\mu\in\Bbb{R},\sigma\gt0$ whose pdf is $f(x\text{;}\theta)={1\over \sigma}e^{-{{(x-\mu)}\over \sigma}}I_{(\mu,\infty)}(x)$ I easily found $(X_{(1)},\bar{...
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Renewal process(arrival distribution) with a specific inter-arrival time distribution [closed]

If an inter-arrival time follows exponential distribution, the arrival distribution becomes Poisson. I understand the derivation of that. However, I want to find an arrival distribution with a ...
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Exponential and Poisson distribution, machine

I have this question where i am unsure how to solve it. X...how often a machine does not work E(X)= 3 per day= 1/8 per hour X-Poisson distributed What is the probability that no machine breaks ...
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Find probability that the equation $x^2+Bx+C=0$ has 2 distinct roots.

Let $B,C$ independent random variables such that $B\sim \operatorname{exp}(\lambda),C\sim U[0,1]$. I have 2 questions about the solution: "We're looking for the probability that $\mathbb{P}(4B^2-4C&...
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Question on proving the inverse of the distribution function of the random variable

We can approximate the conditional distribution by a random variable with density $$ \\p\delta_0(x)+(1-p)\beta e^{-\beta{x}}1_{x>0}dx \ $$ that is to say a weighted mean between a Dirac mass at 0 ...
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probability that there n people in a birth and death process?

What is the general formula of probability that there n people when considering a birth and death process? I researched it in some books, but i just found some questions in which increasing/...
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life expectancy of new computer with exponential distribution

I will appreciate someone to verify my answer for exponential distribution question as I am teaching myself and don't have much confidence. Let X, the number of years a computer works, be a random ...
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1answer
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Convergence in distribution of minimum of IID random variables

I'm stuck on the following problem and could use a hint: Let $Z_1,\ldots,Z_n$ be IID random variables with density $f$. Suppose that $\mathbb{P}(Z_i > 0) = 1$ and that $\lambda = \lim_{x \to 0^{+...
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Solving Exponential Distributions with Preemptive Queueing

A router implements a preemptive priority queueing policy, where high priority packets are served first, and their arrival interrupts low priority packets’ service. If the service is interrupted, the ...
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1answer
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“Following” $\operatorname{exp}(\lambda)$ random variables “sum” to $\operatorname{Poi}(\lambda t)$ random variable

Lifetime of a bulb is distributed $\operatorname{exp}(\lambda)$. When one light bulb is burned we replace it immidietely. Let $N_t$ be the number of bulbs we've used by time $t$. Prove that $N_t \sim \...
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Suppose that $f (x, y) = xe^{−x(y+1)}$, where $0 ≤ x < ∞$, $0 ≤ y < ∞$. Find marginal densities

This question comes from rice 3.14 Suppose that $$f (x, y) = xe^{−x(y+1)}$$ where $0 ≤ x < ∞$, $0 ≤ y < > ∞$ a. Find the marginal densities of X and Y . Are X and Y independent? b. ...
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Hints Lack of Memory implies Exponential Distributiom

Let $X$ be a continuous Random Variable with a continuous Distribution function and with values in $(0,\infty)$, and with a denisty w.r.t the lebesgue measure. I have already been able to prove that ...
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Finding the moment generating function of $\min(Y,1)$

Let $Y\sim\text{Exp}(1)$ be a random variable. I denote the random variable $X$ as $X=\min(Y,1)$. The task is to find the moment generating function of $X$. By simply calculating the probability I ...
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1answer
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probability re: comparing 2 iid exponential random variables

Let $X, Y$ be iid exponential random variables with parameter $1$. Then, what is the probability that $X < Y + 1$? I know how to compute $P(X < Y)$ (from integrating the joint PDF, which is ...
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Probability of a machine working at certain time

This question arises while I am learning Continuous Time Markov Chain : A machine is working for an exponential time with rate $\mu$ before breaking down. The repair time of the machine is ...
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2 Poisson distributions time distribution given one of them occurs first

Suppose that A and B are two independent Poisson distribution with parameter $\lambda_a$ and $\lambda_b$ denoting the number of, say the received emails per hour in two different computers. B is ...
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Convolutions: Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X.

Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X. Solution: $f_T(t) = \int_{-\infty}^{\infty}f_Y(t-x)*f_X(x)dx$ $= \int_{-\infty}^{\infty}1*\lambda e^{-\lambda x}dx$ Integrate ...
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Exponential distribution of 2 independent events (expected value)

X and Y are waiting time between phone calls for company A and B respectively and they are independent from each other. X and Y are exponentially distributed with expected waiting time of 10 min and 8 ...
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Prove that almost surely $\lim\sup_{n\to\infty}\frac{X_n}{\ln n}=\frac{1}{\lambda}$

Let ${X_n}$, n=1 to infinity, be independent random variables distributed $Exp(\lambda)$. Prove that almost surely $$\lim\sup_{n\to\infty}\frac{X_n}{\ln n}=\frac{1}{\lambda}$$ My idea was to look at ...
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Exponential Distruproblem

Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from 4 tigers in the reserve follow an iid exponential distribution with mean 3. If the system has to ...
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The distribution of the ratio of two exponential distributions [duplicate]

I have a question about the pdf of the ratio of two exponential distribution. Suppose that X and Y are independent EXP(1) R.V., what is the pdf of Z:=X/Y? My idea is we need to get the cdf of Z and ...
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1answer
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Limes superior of $X_n>\alpha \ln(n)$

Given the random variable $X_n$, to be exponential distributed with parameter $1$, we define $$A_n:=\left\{X_n>\alpha \ln(n)\right\}.$$ We've shown in the classes that for $\alpha \leq 1$, $A_n$ ...
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1answer
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Order statistic of i.i.d exponential($\lambda$) random variables, $X_{(n, k_n)}$ convergence in probability

Suppose that $X_1,X_2$,....are iid from exponential($\lambda$).For n $\geq$ 1, let $X_{(n,1)}\le X_{(n,2)}\le X_{(n,3)}\le.......\le X_{(n,n)}$ be the order statistics of $X_1,X_2....X_n$. Suppose ...
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Expected value of $e^x$ as $X\sim exp(\lambda)$

Let $X\sim exp(\lambda)$ Calculate the Expected value of $$Y=e^X$$ Is there any elegant way to do it?
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How can I find the mean value of an exponential profile?

assume a surface density profile \Sigma(y)=\Sigma_p e^{-2y}. How can I find a (fixed) mean value to approximate the mentioned profile? Is it true fo find the value ...
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1answer
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PDF of an exponential distribution with varying paramter, lambda

Suppose that the lifetime of a device is exponential with rate λ, but suppose also that the value of λ is not fixed but is itself a random variable that is uniform in the range [a, b) with 0 < a. ...
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1answer
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Convergence in probability: The inverse of the simple mean

I have a question on convergence: I have to prove that $\frac{n}{U_{n}} \longrightarrow 1$ in probability, where $U_{n}=\sum X_{i}$, $X_{i}\sim \mathrm{Exp}(1)$ and because of this, $U_{n}\sim \mathrm{...
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1answer
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Convergence in distribution of a product

I'm kinda lost in this kind of problems, so I apologize if this is to easy. Let $V_n\sim\operatorname{Exp}(n)$. I've already prove that $V_n \rightarrow 1$ in probability, $e^{-V_n}\rightarrow 1$ in ...
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Exponential distribution of decay

A container contains 13 particles, at time $t = 0$. The particles decay independently of each other and the time (unit: minutes) for a given particle's decay is a exponentially distributed random ...
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Jointly Distributed Random Variables

I have a question below I am stuck on. Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the other lifetimes, and that each lifetime has an exponential ...
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Exponential Distribution - ML estimator of λ in τ parametrization

I am stuck with the following problem: We know that $x_1 \cdots x_n$ ~ Ex$(1/\lambda)$ and we are given the sum $\sum_{i=1}^n x_i$. We want to find the maximum likelihood estimate of $\lambda$, as ...
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N exponentially distributed components with mean τ

I am stuck with the following problem. Let the mean life time of a component be exponentially distributed with mean τ. If we switch on all of them at the same time, how is the waiting time until the ...
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Comparing the Exponential Truncated Distribution with the Exponential Distribution

I have the truncated exponential distribution: $$f(x)=\frac{\lambda e^{-\lambda x}}{1-e^{-\lambda k }} \quad \text{ for } 0 <x<k$$ The expected value of this for $k=\frac{\lambda}{5}$ is $$\...
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CDF and PDF of absolute difference of two exponential random variables.

I'm studying probability theory and came across an exercise problem that I would like to request some help with. Here's the problem: Let $X$ and $Y$ be i.i.d. random variables with Expo($1$), and let ...
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1answer
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KL divergence between Exponential and Normal distributions

What is the KL divergence $KL(P \;\Vert\; Q)$ between an Exponential distribution $P = \text{Exp}(\lambda)$ and a Normal distribution $Q = \mathcal N(\mu, \sigma^2)$? I have not found any source for ...
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Proving $\mathbb{P}(\sum_{n=1}^{\infty} S_n = \infty) = 1$ where $S_i \sim \exp(\lambda_i)$ and $\sup_n \lambda_n < \infty$.

I was working through J.R. Norris Markov Chains and got stuck at exercise 2.3.1. The question is the following: Let $S_1, S_2, \ldots$ be independent exponential random variables of parameters $\...
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Independence of two events, $\{S < T\}$ and $\{\min\{S,T\} \geq t\}$ ($S \sim \exp(\alpha), T \sim \exp(\beta) $)

I was working through J.R. Norris Markov Chains and got stuck at exercise 2.3.1. The question is the following: Suppose $S, T$ are independent exponential random variables of parameters $\alpha$ ...
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Expectation when x>5

Exam question was: Suppose $X$ is an exponential random variable with the expectation of $1$, find $E(X|X>5)$. My answer was that first find $$P(X<x | X > 5)$$ Which I got to be $1-e^{5-...
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Normalising a multi exponential probability density function Bayes theorem

I've been trying to normalise the posterior pdf from Bayes' theorem and I just can't get my head around the mathematics. So starting with Bayes theorem itself, given some variable $\theta$ and some ...
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Generating Random variables for Exponential distribution (long series)

I have number series (1,2,3,......100) . I want to get random variable based on exponential increasing probability distribution such that 1 has lowest probability of being chosen and 100 has highest. ...
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Expectation Maximization(EM) for Mixture of Exponential Distributions

I am new to EM algorithm and I'm dealing with the following question of a Mixture of Exponential Distributions. Suppose that the time from when a machine is manufactured to when it fails is ...
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Probability to create $n$ screens with a probability to have a breakdown

We need five successive working stations to produce a screen and the time spent on each of the working station is distributed as an exponential random variable. The average time of each working ...
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Easy integral,density,definite integral

I cannot compute the following integral. I'm confused with upper and lower bounds which are somehow switched from my point of view: $$\int_0^\infty\lambda e^{-\lambda x}dx=1.$$ This is supposed to be ...
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Cramér-Rao Lower Bound for estimator of mean in Exponential distribution

Let $X_{1},...,X_{n}$ be a random sample of size $n\geq3$ from the exponential family with mean $1/\theta$. (1) Find a sufficient statistic $T(X)$ for $\theta$ and write down its density. (2) Obtain ...
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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 ...
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Supply chain modelling

So I have my first probability and statistics course this year, and we're currently learning about the different distributions that can be used to model, for example, supply chains. I was wondering ...