Questions tagged [exponential-distribution]

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

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7
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1answer
27 views

The time it takes for a candle having a lifetime that follows an exponential distribution to go off

We have $5$ candles each having a lifetime which follows an exponential distribution with parameter $\lambda$. We light up each candle at time $t=0$. Assume that $Y$ is the time that it takes for the ...
-4
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0answers
17 views

Adding vs multiplying means and variances of exponential distribution

If X1 and X2 are both random variables with the same exponential distribution, what would the mean and variance be of X1+X2? What would the mean and variance be of 2(X1)? I assumed that both mean and ...
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0answers
13 views

Anderson-Darling test statistic

The Anderson-Darling test statistic is defined as Why is it useful in testing whether an exponential distribution appropriate or not to model the quakes dataset referring to the probability plot and ...
0
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0answers
23 views

Complete and sufficient statistic

Let $X_1\dots X_n$ be iid observations with pdf $f(x|\theta)=e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$ for $-\infty<x<\infty$ and $-\infty<\theta<\infty$. I need to find a complete sufficient ...
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1answer
29 views

Four students are giving presentations

In four sections of a course, running (independently) in parallel, there are four students giving presentations that are each Exponential in length, with expected value of 10 minutes each. How much ...
0
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1answer
28 views

$\mathbb P(X_1 < X_2)$, given that $X_1$ and $X_2$ are both exponentially distributed.

In a tutorial exercise I am asked to determine: $\mathbb P(X_1 < X_2)$, given that $X_1$ and $X_2$ are both independently exponentially distributed with the same parameter $\lambda$. I would be ...
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0answers
15 views

lifespan exponential distribution problem [closed]

I'm preparing myself for the random process test. I do not know how to solve this problem. We have five lamp whose lifespan each follows an exponential distribution with the Landa parameter. We light ...
0
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2answers
30 views

If $X\sim U(0,1)$ then $Y=-\frac{1}{\lambda }\ln(1-X)\sim Exp(\lambda )$

If $X\sim U(0,1)$ then $Y=-\frac{1}{\lambda }\ln(1-X)\sim Exp(\lambda )$ Here is my solution : If $X\sim U(0,1)$ then $f(x)=1$ where $0\leq x\leq 1$. $0<x\Rightarrow -x\leq 0\Rightarrow 1-x\leq 1\...
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1answer
13 views

probability of exponential distribution question

Suppose $X_n$ follows Exponential distribution with parameter $\alpha$. Find P($X_n$ < log(n)*$\epsilon$). I got my answer equals 1 - $\frac{1}{n^(\alpha\epsilon)}$, which is quite different from ...
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0answers
10 views

The Weibull distribution in one-parameter exponential family

I have been given a non-standard Weibull distribution $$f(y,\beta)=\beta \alpha y^{\alpha-1} exp(-\beta y^\alpha)$$ The exercise is to show above distribution is a one-parameter exponential ...
1
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0answers
11 views

Poisson Process timed on other arrivals

I've been struggling on this question for a while. Particularly, part b. My thought process was that it uses 2C1 competing exponentials to end up with that result, although it does not make intuitive ...
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0answers
20 views

Prove that $S_N = \sum_{i=1}^{N} X_{i}$ is exponentially distributed with parameter $p$ [duplicate]

Let $X_{1}, X_{2}, \ldots$ be exponential random variables with parameter $1 .$ Let $N$ be a geometric random variable with parameter $p$. Prove that $S_N = \sum_{i=1}^{N} X_{i}$ is exponentially ...
0
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1answer
24 views

Proving Transformation of Random Variable is Chi-Square with 2 Degrees of Freedom

Let the cumulative distribution function for the random variable $T$ be given by $$P(T \leq t) = \begin{cases} 1 - e^{-\frac{t}{30}} & t \geq 0 \\ 0 & t < 0 \end{cases}$$ Prove that the ...
1
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1answer
28 views

The distribution of an event whose waiting time is exactly k

I am trying to solve the following problem Assume busses arrive with the interval exactly 10 min one after another. At the end of a working day, I take a bus home. Let α be my expected waiting time ...
-1
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1answer
31 views

An argument on exponential distribution

Based on the question I asked here, if we assume that the independent random variables $U$ and $X$ follow the exponential distribution with parameter $\lambda$ and the general distribution $f_X(x)$, ...
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0answers
16 views

Expected completion time from a exponential distribution

I have the following problem: A person do a manual activity during a time that is exponentially distributed with mean "lambda". If the person has been working during a period of length "...
1
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1answer
21 views

Find the joint distribution and covariance with exponential density

Let $X_1, \dots , X_n$ be independently distributed with exponential density $$ f(x) = (2θ)^{−1}e^{−x/2θ}, x \geq 0 $$ and let the ordered $X$’s be denoted by $X_{(1)} \leq X_{(2)} \leq \cdots \leq ...
2
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3answers
33 views

Let $X$~$U(0,5)$ & $Y$ be exponential random variable with with mean $2x$. Find the mean and variance of Y.

Let $X$~$U(0,5)$ & $Y$ be exponential random variable with with mean $2x$. Find the mean and variance of Y. My Working: I know that the pdf of random variable $X$ is given by: $f_X(x)=1/5$ Since $...
2
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2answers
81 views

Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$

Let $X_1,X_2,\ldots,X_n$ be a random sample with $n\geq 2$ from an exponential distribution. $X_{(1)}=\min(X_1,X_2,\ldots,X_n)$. Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$. I was able to find ...
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0answers
24 views

Probability distribution for number of concurrent events

Say we have a type of event that lasts 30 seconds and occurs on average once per second. I know that you can use an exponential distribution to describe time between events, or a Poisson distribution ...
0
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1answer
33 views

Memoryless property: how does $P(X > s + t \mid X > s) = P(X > t), s, t \ge 0$ imply that $F^c(s + t) = F^c(s) F^c(t), s, t \ge 0$?

I am currently studying the textbook Modeling and analysis of stochastic systems, third edition, by Kulkarni. Chapter 5.1.1 Memoryless Property says the following: We begin with the definition of the ...
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0answers
27 views

Conditional expectation of independent exponential random variables [closed]

Let $X$ and $Y$ be independent exponential random variables of parameter $\lambda > 0$. We define $Z=min\{X,Y\}$ and $W=min\{X,Y\}$. Show the following statements: (1) $E[Z|X]=\frac1\lambda (1-e^{-\...
0
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1answer
39 views

Finding $E(X\cdot\mathbb{1}_{X\ge t}\mid Y_{t})$ where $X\sim\text{Exp}(\lambda)$ and $Y_{t}=\min(t,X)$

Suppose $X\sim \text{Exp}(\lambda)$ and $Y_{t}=\min(t,X)$. Show that $$E(X\cdot\mathbb{1}_{X\ge t}\mid Y_{t})=\left(t+\frac{1}{\lambda}\right)\cdot\mathbb{1}_{X\ge t}$$ I know that since $X\cdot\...
1
vote
1answer
60 views

$\tilde{F}_X(s) = E\left( e^{-sX} \right) = \int_0^\infty e^{-sx} f_X(x) \ dx = \frac{\lambda}{\lambda + s} \,, \ \text{Re}(s) > - \lambda$?

I am currently studying the textbook Modeling and Analysis of Stochastic Systems, third edition, by Kulkarni. Chapter 5.1 Exponential Distributions says the following: The probability density ...
0
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1answer
40 views

Moment estimator and its asymptotic distribution for exponential distribution

I have this question: We let $X$ and $Y$ be independent exponentially distributed random variables with $E(X)=\beta$ and $E(Y)=\frac{1}{\beta}$,$\beta>1$. We let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be a ...
0
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1answer
24 views

understanding the convolution of random variable formula

Consider summing two iid exponential r.v. We know for a fact that this is Gamma distribution with $\alpha = 2, \beta = \theta$. However, when using the convolution formula $Z=X+Y$, we have $$ f(z)= \...
0
votes
3answers
62 views

$Z_i=X_i-Y_i$ with $X_i$ and $Y_i$ independent and exponentially distributed

I got this question: We consider a sample of $Z_1,...,Z_n$ of independent and identically distributed random variables where $Z_i=X_i-Y_i$ with $X_i$ and $Y_i$ independent and exponentially ...
1
vote
1answer
132 views

What is the cumulative distribution function of T?

Define 3 independent random variables: X the time until your friend Alex calls, with an average waiting time for his call 1/λ (λ > 0) Y the time until your friend Jessica calls, with an average ...
0
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0answers
26 views

Most Powerful test for deciding distribution

Let $X_1,...,X_n$ be iid distribution function of $F(x)$. I want to test whether $F$ is exponential or Weibull. This means that either $F(x)=1-exp(-x), x>0$ (exponential) or $F(x)=1-exp(-x^{\theta})...
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3answers
36 views

Probability distribution of total life time of a machine with two parts in parallel system

Two identical components having lifetimes $A$ and $B$ are connected in parallel in a system .Suppose the distributions of $A$ and $B$ independently follow exponential with mean $\frac 1a, a>0$. ...
0
votes
1answer
37 views

minimum of two exponential random variables over an interval $[0,T]$ instead of $[0,\infty]$

Let $X_i$, $X_j$ be two independent exponentially distributed random variables with rate parameters $\lambda_i$ and $\lambda_j$. Then we know that $\min\left\{X_i, X_j \right\}$ is also ...
0
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0answers
19 views

Probability of Ruin of Insurance Model with Exponential payoffs

I'm having trouble proving a result I think should be true for the insurance payout model. Say you have $U_n = x + c*n - X_n$ where $X_n = X_{n-1} + Y_n$ and the $Y_i$ are all iid $Y_i \sim \text{Exp}(...
1
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0answers
24 views

What is the probability that I would wait more than 5min for the bus?

I'm waiting at a bus stop, I can take the bus 34, which I need to wait for, on average, 10 min, and the bus 45, which I need to wait for, on average 4 minutes. The times of arrival of the buses are ...
0
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1answer
41 views

Concluding that MLEs for exponential distribution parameters are biased and then unbiasing them

I have the i.i.d. exponential random variables $X_1, \dots, X_n$ with the density functions $$f(x; \sigma, \tau)= \begin{cases} \dfrac{1}{\sigma} e^{-(x - \tau)/\sigma} &\text{if}\, x\geq \tau\\ ...
1
vote
1answer
37 views

What is the reasoning involved in finding the sufficient statistic for the shifted exponential distribution?

I have the i.i.d. exponential random variables $X_1, \dots, X_n$ with the density functions $$f(x; \sigma, \tau)= \begin{cases} \dfrac{1}{\sigma} e^{-(x - \tau)/\sigma} &\text{if}\, x\geq \tau\\ ...
0
votes
1answer
116 views

Find the distribution function of $V = U-X$ when $U$ follows the exponential distribution and $X$ follows the general distribution.

Assume that the random variable $U$ follows the exponential distribution with parameter $\lambda$; also, suppose that random variable $X$ follows the general distribution $f_X(x)$. For $U>X>0$, ...
0
votes
1answer
13 views

exponential distribution and the gamma function

I don't find the relationship between the expected value (theoretical) of an exponential distributed variable and the gamma function. I work on the paper Moments of the Log ACD model from Luc Bauwens(...
0
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1answer
60 views

MLEs for shifted exponential distribution: What am I doing wrong and how do I calculate them?

I have the i.i.d. exponential random variables $X_1, \dots, X_n$ with the density functions $$f(x; \sigma, \tau)= \begin{cases} \dfrac{1}{\sigma} e^{-(x - \tau)/\sigma} &\text{if}\, x\geq \tau\\ ...
0
votes
1answer
47 views

Poisson Process and Exponential Random Variables

Suppose we have a planetary system with $m$ planets, orbiting a star. Suppose the distance from the star to each planet is independent exponential random variable with rate $\mu$. I want to find the ...
2
votes
2answers
38 views

Obtain $P\left( \sum_{i=1}^{k} Y_{i}<X<\sum_{i=1}^{k+1} Y_{i}\right)$ for $X$ and $Y_{i}$ independent exponential RVS

Given that $X$ is an exponential random variable with parameter $\lambda$ and $ Y_{i}$ are independent and identically distributed exponential random variables with parameter $\beta$. Given that X and ...
3
votes
1answer
74 views

What is the reasoning used here to determine the UMVUE?

Let $X_1, \dots, X_n$ denote a random sample from the PDF $$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{...
1
vote
2answers
85 views

Calculating method of moments estimators for exponential random variables

I'm trying to find the method of moment estimators for $\sigma$ and $\tau$. I have the i.i.d. exponential random variables $X_1, \dots, X_n$ with the density functions $$f(x; \sigma, \tau)= \begin{...
1
vote
0answers
29 views

Expected value of exponential distribution with integration by parts (non-textbook way)

I try to calculate expected value of exponential distribution with integration by parts without success. I know that the text book way to do it so that first take lambda out of the integral and then ...
0
votes
1answer
75 views

Hypothesis Testing for Exponential Distribution Mean

I'm having some trouble answering the following question: Suppose that the lifetime of batteries produced using certain materials is exponentially distributed with parameter $\lambda$ (density ...
0
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0answers
8 views

Enforcing conditions on cumulative truncated exponential distribution

The CDF for an exponential distribution of rate $\lambda$ truncated at T is $F(t) = \frac{1-e^{-\lambda t}}{1-e^{-\lambda T}}$. (for $t<T$, else 0). I would like to determine $\lambda$ and $T$ such ...
0
votes
1answer
41 views

The mean of exponential distributions

I would like to ask the following question. Let's say we have two independent variables of x and y, and both are exponentially distributed with mean $\mu_1, \mu_2$. And let $S=x+y$. May I know whether ...
0
votes
1answer
28 views

What form would the formula describing this behaviour have? [closed]

exponential(?) behaviour of my parameter as a function of temperature. I have a simple question relating to the above figure. I want to fit (leastsquare) a line trough this data using a natural ...
0
votes
1answer
36 views

Given ratio of consecutive numbers to arrive at fraction of total number [closed]

Given The ratio of number of oscillators in their (n+1)th quantum state of excitation to the number in their nth quantum state is $$N_n+1/N_n = \exp(-\hbar\omega/K_bT)$$ Thus the fraction of the total ...
0
votes
1answer
109 views

Find the Maximum Likelihood Estimator for the Given Equation [closed]

How should I go about determining the maximum likelihood estimator for beta (part B)? Also, what is the distribution this question is asking for, and how can I calculate the probability that the ...
0
votes
3answers
59 views

X ∼ Exp(1) and Y ∼ uniform(0, 1) are independent. Compute $E(e^{-XY^2})$.

X ∼ Exp(1) and Y ∼ uniform(0, 1) are independent. Compute $E(e^{-XY^2})$. I am having trouble answering this question, I am not sure if I should take advantage of moment generating functions or ...

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