# Questions tagged [exponential-distribution]

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

672 questions
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### Devising a hypothesis test for machine failure rate

I'm trying to devise a hypothesis test for failure rate data of machines. The gist is that there are some machines in a factory that run all the time. They fail from time to time and are promptly ...
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### Poisson from Exponential

The probability distribution of time between events following a Poisson point process with parameter $\lambda$ is an Exponential Distribution with parameter $\lambda$. The proof from Poisson to ...
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### Computing CDF from an exponential r.v. X and Bernoulli r.v. Z

I need to compute the CDF of Y=ZX and i am struggling to compute this when Y=ZX Information: X is an exponential random variable with parameter 1 Z a Bernouilli random variable taking its values ...
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### Asymptotic distribution of MLE of joint exponential distributions

Given $X_1,...,X_n \sim \text{Exp}(\theta)$ and $Y_1,...,Y_n \sim \text{Exp}(\frac{1}{\theta})$, where $\theta>0$ and have the same $\theta$ in both distributions. $X$ and $Y$ are independent. I ...
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### MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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### How can I find the expected value of this variable?

The initial cost of a machine is 3\$. The life-time, T, of that machine has an exponential distribution with an expected value equal to 3 years. The maker wants to offer a warranty that pays 3\$ if ...
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### Characteristic function of exponential distributed random variable

Given: $$f_X(x) = \lambda e^{-\lambda x},\; x\in X$$ Wanted: The corresponding characteristic function $\phi(ju)$. \begin{align} \phi(ju)&=\mathbb{E}(e^{j^2ux})\\ &= \lambda \int^{\infty}...
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### Joint Exponential Probability Problem

I have a question from A First Course in Probability by Sheldon Ross. Question: Consider two components and three types of shocks. A type 1 shock causes component 1 to fail, a type 2 shock causes ...
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### Probability generating function of exponential distribution

The exponential distribution is given by: $$PDF: \lambda e^{\lambda x}$$ And the formula for probability generating function is given by: $$G(z) = \sum_{x=0}^\infty p(x)z^x$$ where $p(x)$ is a ...
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### 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 ...
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### Find the pdf , distribution function of $X$ and $E[(X-2)^2]$

I 'll be very grateful if you can help me , here is the question : When a person sends an email, the probability that there is an attachment is 0.5. If there is an attachment then the size of the ...
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### Exponential distribution MLE with lifetime and frequency table

\begin{array}{c|c} \hline \text{Lifetime (months)} & \text{Observed frequency} \\ \hline 0-2 & 50 \\ \hline 2-4 & 35 \\ \hline 4-6 & 25 \\ \hline 6-8 & 15 \\ \hline 8-10 & 5 ...
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### Hypothesis testing for the lifetime of bulbs using subsystems

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean λ. To test the null hypothesis H0 : λ = 1000 against the alternative ...
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### Introduction of shape parameters in the formulation of probability distribution

I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the ...
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### a constant $k$ of a Weibull Distribution

I am supposed to show the constant of a Weibull Distribution, $k$, is a product of $\alpha$ and $\beta$. I know $\alpha = \frac{1}{\theta}$ ($\theta$ being the probability of a success) for the ...
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### PDF of the time at which the second raindrops hits

Raindrops hit you at a rate or 1 raindrop per second, what is the PDF of the time at which the second raindrop hits you? Clearly, we have to use exponential random variable. Also we are asked to use ...
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### Poisson Process from independent non-identical exponential RVs

I know, I can define a Poisson Process using a sequence of i.i.d. exponential random variables, i.e. let $\tau_1, \tau_2, \tau_3, ... \sim \mathrm{Exp}(\lambda)$, then $T_i = \sum_{j=1}^I \tau_j$ are ...
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### Why isn't expected time to leave a system with exponential service times simply the shortest expected service time times two?

This is from exercise 25, chapter 5, in Ross' Introduction to Probability Models, 11th ed. It goes as follows: Customers can be served by any of three servers, where the service times of server $i$ ...
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### If $X$ is an exponentially distributed variable with mean $\lambda$, $Y=−3\ln(X)$ has Gumbel distribution?

Let X be a random variable which follows an exponential distribution with parameter $\lambda$ ($\lambda>0$), find the distribution of the random variable $Y = −3\ln(X)$. So this is my answer for ...
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### Expectation of number of broken machines by time t

$n$ machines in total (starting working at the same time), working time of each is i.i.d. $Exp(\lambda)$. How to calculate the expectation of the number of broken machines at time $t$? (If broken, ...
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### Probability between two events following exponential distribution

Two independent events A and B follow exponential distribution with parameter L: f(t)=Le^(-Lt) for t>=0. If X is the time where A occurs and Y the time where B occurs, calculate the probability P[X>=2*...
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### Probability of falling meteor (Poisson process)

Let's say, that we know, that in some period of year there are really common sightings of meteors with an average of 100 meteors per hour. What's the chance of no meteor seen in 5 minutes? I was ...
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### How to decelerate from velocity v to stop time t over distance d?

I'd be grateful for some help with this problem I am trying to solve. Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
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### How to distort a sigmoid / logistic function?

Please help. I need to move an object such that its distance / time profile resembles a sigmoid curve but in a non-symmetrical manner. 1- Let's say that I need to move 800cm in 3 seconds. How do I ...
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### Let $X$ be a non-negative random variable, and $A_i$ be the event that $X$ is between $[i - 1, i)$.

Let X be a non-negative random variable, and $A_i$ be the event that $i-1 \leq X < i$. (a) Show that $\sum_{i=1}^{\infty}(i-1)I_{Ai} \leq X < \sum_{i=1}^{\infty}iI_{A_i}$, where $I_{A_i}$ = 1 ...
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### Lambda and mean of sum of 2 independent exponential random variable

I have a restaurant with service time being exponentially distributed. Let's say the food-serving time has a mean of 30 minutes, and the checkout time has a mean of 10 minutes. Would $\lambda$ in ...
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### Poisson Process with Expected Time (Stochastic Processes)

Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 30 minutes to clean the kitchen. Kelly ...
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### Difference of two exponential distribution

Let $X\sim Exp(\lambda_1)$ and $Y\sim Exp(\lambda_2)$, with $Z = X - Y$. I am trying to find the pdf of Z, i.e. $f_Z(z)$. Here is what I have got: \begin{align*} f_Z(z) &= \int_0^{z}...