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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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How to find the MLE of these parameters given distribution?

Let $X$ and $Y$ be independent exponential random variables, with $$f(x\mid\lambda)=\frac{1}{\lambda}\exp{\left(-\frac{x}{\lambda}\right)},\,x>0\,, \qquad f(y\mid\mu)=\frac{1}{\mu}\exp{\left(-\...
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1answer
29 views

Confused about how to show independent random variable $Y$ has the Poisson distribution with parameter $t\lambda$

Assume there are N independent exponential random variable $(X_1, X_2,..., X_N)$ with parameter $\lambda$. Fix a real number $t > 0$. Let Y be the largest $N$ so that $X_1 + X_2 + \ldots + X_N \...
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0answers
18 views

Exponential Test

Given a collection of data points, without any other information I want to test for an exponential distribution. In a different case, I tested for normality using an Accord implementation of the ...
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25 views

Poisson with Exponentially Distributed Parameter

I am doing a review for the test, and I have found myself really struggling with the following question: Prove using generating functions that if $$V \sim \mathrm{Poi}(\theta),~\theta \sim \mathrm{...
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Question regarding Erlang distribution with shape parameter $k=2$ [on hold]

Let $X_i$ be i.i.d. random variables with the CDF $$ F_{x_i}(x_i)=1-e^{-λx_i},\quad \quad x\geq 0. $$ Define $$ X= X_1+X_2+....+X_k\ . $$ A theorem states that $X$ obeys the Erlang-k distribution ...
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1answer
26 views

Let $x_1=\exp(\lambda_1),x_2=\exp(\lambda_2)$, $x_1, x_2$ are independent show $P(x_1<x_2)=\frac{\lambda_1}{\lambda_1+\lambda_2}$ [on hold]

Let $x_1=\exp(\lambda_1),x_2=\exp(\lambda_2)$, and $x_1, x_2$ are independent random variables. Show that $P(x_1<x_2)=\dfrac{\lambda_1}{\lambda_1+\lambda_2}$
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1answer
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conditional expectation of gamma distribution with alpha = 1 [on hold]

$X_i$ are exponential(\lamda) distribution and identically independent distribution. Y = $\sum_{i=1}^n$$X_i$ $X_i$ is an unbaised estimator of \lamda. Y is a sufficient estiamtor of \lamda. solve ...
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13 views

PDF of the time at which the second raindrops hits

If we look at the question: "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 ...
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12 views

K-moment for exponential distribution [closed]

How can I find k-moment for exponential distribution using chatacteristic function?
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1answer
42 views

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}...
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21 views

Sum of 2 exponential distribution with different parameters

Let $X\sim Exp(\lambda_1)$ and $Y\sim Exp(\lambda_2)$, I am trying to find the distribution of $Z = X+Y$. I understand that $f_z(z)=\frac{\lambda_1 \lambda_2}{\lambda_2-\lambda_1} \left(\exp[-\...
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1answer
35 views

Probability minimum is “reached”

Let $(X_i)_{i=1,\dots,n}$ be a finite sequence of random variables such that $X_i\sim\mathcal{E}(\lambda_i).$ We can prove that $Y:=\min_{1\le i\le n}X_i\sim\mathcal{E}(\lambda=\sum_{i=1}^n \lambda_i)...
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1answer
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Density fuction of Exponential distribution [closed]

Suppose that X1, X2,...,Xn are a random sample from the exponential distribution with mean b. Find the density function of the largest observation, Y = max Xi
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1answer
31 views

If $X\thicksim \text{Exp}(\lambda)$ and $Y\thicksim\text{Geom}(p)$ are independent, find $\mathbb P(\lfloor X\rfloor=Y)$

Suppose that $X$ and $Y$ are independent, $X\thicksim \text{Exp}(\lambda)$ and $Y\thicksim\text{Geom}(p).$ Let $\lfloor x\rfloor$ be the floor function (largest interger which is at most $x$). Find $\...
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Probability of Ruin at the first claim

The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution. Assume the initial surplus is $U$, and the ...
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1answer
37 views

Probability that lightbulb stops working in odd year

I have a lightbulb that has an exponential lifetime distribution with mean $\mu$ months. So if I construct a pdf $f(x)$ and cdf $F(x)$ with parameter $\lambda$, and since $E[X] = \frac{1}{\lambda}$, \...
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1answer
16 views

Find the p value following the exponential distribution $\mu=3$

I want to find the $p$-value (manually) of the following Hypothesis testing. $$H_0:\mu\leq 3 \quad \text{vs} \quad H_1:\mu >3$$ The main thing I know is that $$P(\mathrm{Re}\,j \mid \mu \leq 3)=...
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1answer
42 views

Proof of $P(X<Y)$

Assume that $X$ is $Exp(\lambda)$ distributed and $Y$ is $Exp(\mu)$, and they are independent. I want to know how I can calculate $P(X<Y)$. I don't understand why $$ P(X < Y)=\int_{-\infty}^\...
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1answer
26 views

Derivation of the expectation of exponential random variable

I am following some course notes for the expectation of an exponential random variable, $X \sim \text{Expon}(\mu)$. I believe this is a correct derivation: $$ \begin{align} \mathbb{E}[X] &= \int_{...
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1answer
46 views

Find the probability that $P(X_1\leq \alpha \cap X_2\leq X_1)$ and $P(X_1+X_3 \leq \alpha \cap X_1< X_2)$

Let $X_1, X_2$ and $X_3$ three positive independent random variables. The PDF of and CDF of $X_i$ are $f_{X_i}(x)$ and $F_{X_i}(x)$ respectively. For example, for exponential random variables we ...
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How to calculate realization of a random process?

I am newly learner of subject of stochastic processes and my mind is full of questions. Hopefully I can ask one of them in a correct way. Suppose that X is an random variable which follows ...
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0answers
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Probability $P\{\min{X_1,X_2}\leq X_3\}$?

Let $X_1, X_2$ and $X_3$ exponential random variables with the same parameter $\beta$? The PDF and CDF of $X_i$ are $$ f_{X_i}(x)=\beta e^{-x \beta}, $$ $$ F_{X_i}(x)=1-e^{-x \beta}. $$ The PDF and ...
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1answer
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Proof on how to sample from a truncated exponential distribution

I understand that if i want a sample from an exponential distribution left truncated at a, i can just take a sample from a regular exponential distribution and add the value of a to every single ...
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1answer
33 views

The average time before a person find their group

Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\...
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1answer
64 views

Show that if P{$t_0 \le t \le t_0 + t_1 | t \ge t_0$} = P{$t \le t_1$}, then P{$t \le t_1$} = $1 - e^{ct_1}$

I am attempting Question 2-13 from 'Probability, Random Variables and Stochastic Processes' (p.36) by Athanasios Papoulis. The space S is the set of all positive numbers t. Show that if P{$t_0 \le ...
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1answer
37 views

Sequence of i.i.d $\exp(\lambda)$ distributed rv´s. Show that it converges against a rv Z

Let $ X_n $ be a sequence of iid $ \exp(\lambda)- $ distributed rv´s with $ \lambda > 0 $. Show that $ ( \max_{1\leq i\leq n} X_i ) - \lambda^{-1} \log(n) \overset{D}\rightarrow Z $ where $Z$ is ...
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1answer
42 views

Bus arrival times and minimum of exponential random variables

I came across a question that is supposed to show us how the properties of the exponential distribution can be used. I know and have shown that $$P(X_i<min\{ X_1,\dots,X_n\})=\dfrac{\lambda_i}{\...
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Confidence intervall for Rayleigh distribution's parameter?

I manage to estimate $\sigma$ in the Rayleigh distribution, but how do I get a correct confidence interval for it? The Rayleigh distribution is $$f_X(x)=\frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}}.$...
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1answer
48 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
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Probability of (exponential) arrival in between two (exponential) events

Suppose $\tau_1, \tau_2, \tau_3$ are arrival times and all follow an exponential distribution with parameter $\lambda_1, \lambda_2, \lambda_3$, respectively. I fail in deriving the probability of ...
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1answer
32 views

How do I find P(X > x + y | X > x), y > 0 when X is an exponential RV

I'm struggling to see how I would manipulate the PDF so that I could find the conditional probability? X is an exponential RV with $\lambda>0$ and the pdf=$\lambda e^{-\lambda x}$
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How do I find $E\lfloor X \rfloor$ when $X \sim \text{exp}(2)$?

I am trying to learn from my mistakes, and faced the following problem: Let $X \sim \exp(2)$ and $Y=\lfloor X \rfloor$, compute $E[Y]$. Well my false attempt was: First compute the PDF of Y: $ ...
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Density Function of Series Connection

The random lifetime of a certain electronic component is given by- $ f_Y(y) = \dfrac{1}{a} e^{-(y/a)}, y>0 $. Two such components are connected in series. The system fails when the first component ...
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Calculate the confidence interval of parameter of exponential distribution with summarySE in R?

I am trying to calculate the confidence interval for a set of data with the assumption they follow Exp dist. To achieve this, I am merging this with this in R, but does not work as I am not very ...
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1answer
37 views

If $X\sim\text{Exp}(\lambda)$, then $\textbf{E}(X^{n}) = \displaystyle n\textbf{E}(X^{n-1})/\lambda$

If $X\sim\text{Exp}(\lambda)$, then $\textbf{E}(X^{n}) = \displaystyle n\textbf{E}(X^{n-1})/\lambda$ MY SOLUTION According to definition of $k$-th moment, we have \begin{align*} \textbf{E}(X^{n}) &...
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2answers
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Find $\mathbb{E}(h(X) \mid U)$ where $h$ is measurable, $X, Y$ are independent and $U = \max(X,Y)$

Find $\mathbb{E}(h(X) \mid U)$ where $h$ is measurable, $X, Y$ are independent and $U = \max(X,Y)$. $X$, $Y$ follow both an exponential distribution with parameter $\lambda = 1$. I couldn't find a ...
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2answers
28 views

Exponential distribution problem finding probabilities

If the number of minutes it takes a service station attendant to balance a tire is a random variable having an exponential distribution with the parameter $\lambda = 0.2$, what are the probabilities ...
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2answers
52 views

What is the expected delay between the grey buses?

Four yellow buses and two grey buses that could be in any order (with equal probability) are traveling together with the probability of a delay of at least $\mathit{t}$ seconds between any two ...
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1answer
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Exponential Distribution Lifetime of a Lightbulb - Is my solution correct?

Can someone please check if I tackled this question correctly? I don't have answers to refer to. Thank you in advance! The lifetime of a lightbulb expressed in days is exponentially distributed with ...
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1answer
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Equivalence between exponential clocks that ring in different ways

Could someone explain to me why the following facts are equivalent. 1) I have a clock that rings with exponential distribution of parameter $1$. When the click rings a transition to the state $s_1$ ...
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2answers
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CDF of $Z=X_1+\max\{X_2,\,X_3\}$

Let $X_1, X_2$ and $X_3$ be expontial random varibles with paramtre $\beta_1, \beta_2$ and $\beta_3$ respectivly. The PDF and CDF of $X_i$ for $x\geq 0$ are $f_{X_i}(x)$ and $F_{X_i}(x)$ given by \...
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1answer
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Computing pmf and cdf for a function of an exponential random variable

I'm a little stuck on this one due to the nature of the function. Here is the question: $\mathit{T}$ is a $\lambda$ = 1 exponential random variable and $\mathit{f(x)= \lfloor x\rfloor}$ (largest ...
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2answers
113 views

Let $X\sim Poi(10) $ and $Y \sim exp(\frac{1}{10})$ independent. Why $P(X+Y\leq\frac{3}{2}) = P(X=0, Y \leq \frac{3}{2}) + P(X=1,Y\leq \frac {1}{2})$?

Let $X\sim Poi(10) $ and $Y \sim exp(\frac{1}{10})$ independent random raviables I would like to compute: $P(X+Y\leq\frac{3}{2})$ So what I did, which is probably wrong, is the following: Let $X=k$ ...
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0answers
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Show that $P(N \geq n) = (1-e^{-\lambda})^n/ \lambda^n $

The lifetime $X$ (in days) of a device has an exponential distribution with parameter $\lambda.$ Moreover, the fraction of time which the device is used each day has a uniform distribution over the ...
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1answer
32 views

Find the pdf of Z=X/(1+Y)

Been trying to solve this problem for some time now. Any help would be appreciated. X and Y are independent R.V's with distribution exp(a) each. I'm asked to find the pdf of Z with: $$Z=\frac{X}{1+...
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1answer
41 views

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family. I know that the beta distribution is $f(x; \alpha, \beta)={1\over B(\alpha, \beta)}x^{\...
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3answers
103 views

Conditional expectation of minimum of exponential random variables

Let $X_1$ and $X_2$ be independent exponentially distributed random variables with parameter $\theta > 0$. I want to compute $\mathbb E[X_1 \wedge X_2 | X_1]$, where $X_1 \wedge X_2 := \min(X_1, ...
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0answers
29 views

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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1answer
30 views

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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1answer
25 views

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 $X_i$ ...