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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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Let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$.

Let $X$ and $Y$ be exponentially distributed random variables with parameter $1$ and let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$. We ...
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Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.

Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
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How to prove that geometric distributions converge to an exponential distribution?

How to prove that geometric distributions converge to an exponential distribution? To solve this, I am trying to define an indexing $n$/$m$ and to send $m$ to infinity, but I get zero, not some ...
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1answer
177 views

Expected sum of exponential variables until two of them sum to a threshold

In answering finding Expected Value for a system with N events all having exponential distribution, I somehow missed the fact that the OP wanted the expected value conditional on the number of events. ...
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2answers
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What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} \...
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On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will ...
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memoryless property of exponential distributions with random variables

It is true that $P(X>t+s|X>t)=P(X>s)$ for certain values $t$ and $s$. However, how can I show that this still holds if: $T$ is a continuous random variable. That is $P(X>T+s|X>T)=P(X&...
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1answer
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Link between Poisson and Exponential distribution

From my set of notes: "If a Poisson distribution has an average rate of $r$, then the waiting time is exponential with mean $\frac{1}{r}$. When talking about the Poisson distribution we'd be inclined ...
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1answer
223 views

finding Expected Value for a system with N events all having exponential distribution

We have a system in which events happen one after each other. The time interval between each two events shown by random variable $t_i$. So, the time interval between the first and the second events is ...
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202 views

Joint density for exponential distribution

Let $X_1$ and $X_2$ be independent random variables each having a exponential distribution with mean $\lambda = 1$. (a) Find the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$. (b) Get the ...
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Memoryless property for Poisson process

Suppose that arrivals of a certain Poisson process occur once every $4$ seconds on average. Given that there are no arrivals during the first $10$ seconds, what is the probability that there will be 6 ...
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1answer
342 views

Partial sum of order statistics of exponential r.v.'s and $\chi^2$

Suppose $X_i \sim Exp(\frac{1}{\lambda}), i = 1,\cdots,n$, where $f(x) = I_{(0,\infty)}\frac{1}{\lambda}e^{-\frac{x}{\lambda}}$ is the p.d.f. of $X_i$'s. And we have a positive integer $r$, and the ...
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Distribution of sum of exponential variables with different parameters

We have $k$ independent random variables with exponential distribution ($T_1, T_2, \ldots , T_k$), parameters of random variables are ($\lambda,\frac{\lambda}{2},\frac{\lambda}{3},\ldots,\frac{\lambda}...
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1answer
736 views

Uniqueness of memoryless property

How does one prove that the unique continuous distribution with the memoryless property is the exponential distribution? i.e. Suppose we know that a continuous random variable $X$ satisfies $$P\{X &...
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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 ...
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2answers
241 views

Expectation of maximum of arithmetic means of i.i.d. exponential random variables

Given the sequence $(X_n), n=1,2,... $, of iid exponential random variables with parameter $1$, define: $$ M_n := \max \left\{ X_1, \frac{X_1+X_2}{2}, ...,\frac{X_1+\dots+X_n}{n} \right\} $$ I want ...
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1answer
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Confused as to How I should Interpret the Exponential Distribution and its Probability Density Function

I'm confused as to how I should be interpreting the exponential distribution and its probability density function (PDF). I'm specifically referring to the fact that it peaks close to $x = 0$ and then ...
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2answers
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Distribution function of $ X(\omega):=\frac{1}{\lambda} \ln \frac{1}{1-\omega},$ in Lebesgue measure

I am having trouble in finding the right approach to this exercise: Define the probability space $(\Omega, \mathcal{A},P) =((0,1), \mathcal{B}, \mu)$ with $\mu$ as the Lebesgue measure and $ \...
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1answer
163 views

Sum of exponential random variables over their indices

Let $\ X_1,X_2,...\ $ be i.i.d. exponential random variables. For $\ n=1,2,...\ $consider the random variables $\ Y_n= \max\{X_1,...,X_n\} \ $ and $\ U_n=\sum_{i=1}^{n}X_i/i \ $. Show that for each n,...
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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 ...
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2answers
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Intuition behind $E(X-t \; |\; X>t) = E(X)=\frac{1}{\lambda }$ in exponential distribution

Let $X$ be a an exponentially distributed random variable with rate $\lambda $ and let $t$ be a constant. It was shown to me that $$E(X-t \; |\; X>t) = E(X)=\frac{1}{\lambda }$$ However, the ...
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Probability exponential random variable smallest among others.

For $X_1,\dots,X_n$ exponential random variables with mean $E(X_i)=\mu_i$. Now I want to calculate the probability that $X_i$ is the smallest among $X_1,\dots,X_n$. Therefore I am trying to calculate ...
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1answer
49 views

$P(T_{1}<T_{2})=\frac{\omega_{1}}{\omega_{1}+\omega_{2}}$

Supposing that $ T_{1} $ and $T_{2}$ are independent and exponentially distributed, with parameters $\omega_{1} $ and $\omega_{2}$ respectively. Then, $$P(T_{1}<T_{2})=\frac{\omega_{1}}{\omega_{1}+...
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1answer
141 views

Random sample generated for i.i.d variables [closed]

My attempt: Since $f(y;a) = \frac{1}{2a}exp(\frac{-y}{2a})$ for $y>0$, and $f(y;a) = \frac{1}{2a}exp(\frac{y}{2a})$ for $y<0$, it is easy to see $A_n$ is a sufficient statistics for a family $T$ ...
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4answers
366 views

Statistics: Deriving a Joint Probability Function From a Definition of Other PDF's

Here's a particular question I'm trying to understand from the lecture notes. It says: Assume that $Y$ denotes the number of bacteria per cubic centimeter in a particular liquid and that $Y$ has ...
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1answer
117 views

How to find conditional expectation and variance.

If I have an exponential random variable $X$ with mean $\theta =2$, and I need to find the mean and variance of $X$, given that $X<3$, I started with $$E[X|X<3]=\int_{0}^{3}[1-\frac{F(x)}{F(3)}]...
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1answer
108 views

Distribution of median of three exponential observations

I just had a question in my exam that I couldn't answer, I bugged. You're given $X = \text{Median}(Y_1, Y_2, Y_3).$ $X$ follows an exponential distribution and you're asked to find the pdf of $X.$ ...
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1answer
356 views

Density of sum of independent exponential random variables

I have two independent exponential RV's $X$ and $Y$ with parameters $\lambda_1$ and $\lambda_2$ respectively. I want to find the density of $X + Y$. I've tried using the convolution formula and got ...
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1answer
1k views

inter arrival time ( continuous exponential distribution)

The customers arrive at store with exponentially distributed interarrival times with a mean of 30 minutes. What is the probability that interarrival time between two successive customers is 2 hours ...
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0answers
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Expected sum of exponential variables respecting a threshold

This question continues the previously asked question which is solved by @joriki. As you can see in the figure, there is cahce which I refered as system in question. Data item $i$ is requested by ...
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2answers
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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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Derived distribution: PDF of $\sin^{2}(x)$ [duplicate]

I want to find the PDF of $Y = \sin^2(X)$, where $X ∼ \text{Exponential}(\lambda)$ is a random variable. I thought i had solved this correctly by simply finding the inverse $h(Y) = X$ and taking the ...
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1answer
2k views

Lifetime of system modeled using exponential distributions

I am reading Probability and Statistics for Engineering and the Sciences. Exercise 15, Chapter 5 says: Consider a system consisting of three components as pictured. The system will continue to ...
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1answer
220 views

exponential continuous distribution

A box of candy contains 24 bars. The time between demands for these candy bars is exponentially distributed with a mean of 10 minutes. What is the probability that a box of candy bars opened at 8:00 ...
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1answer
117 views

what is the Poisson point process formula in this exercice and generally

i've seen in some tutorials and courses that the poisson process formula is : $P(X=n) = e^\left(-\lambda\right) \frac{\lambda ^n}{n!}$ but facing some problems using this formula i did some ...
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1answer
626 views

Independence of spacing of order statistics of exponential distribution

Let $X_{1}$,$X_{2}$ and $X_{3}$ be a random sample from exponential distribution with density function - $$f(x) = \lambda e^{-\lambda x}$$ $x>0$ and $\lambda>0$ Show that $X_{(1)}$,$X_{(2)}$-$...
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1answer
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twelve orders being sent in an hour

Restaurant is sending orders every 5 minutes on average. Question: From 15:00 until 15:05, two orders were ordered. What's the probability that the next order will be ordered until 15:10 ? I can ...