Questions tagged [exponential-distribution]
To be used for questions on using, finding, or otherwise relating to Exponential Distributions.
1,476
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Variance of Poisson and Exponential distribution [closed]
I have encountered a problem with the variance of Poisson and exponential distribution.
Suppose there is an accident that follows a Poisson distribution with an occurrence rate of $\lambda$ per hour. ...
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0
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Sum of Cumulative Max of Exponentially Distributed Variables
Let $X_1, X_2, ..., X_n$ be independent, identically and exponentially distributed random variables, $P(x) = k \exp(-k x)$. Define $Y$ as the sum of the sequence of cumulative maxima:
$Y = X_1 + \max(...
- 352
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How to apply queuing theory to find the long run proportion of customers who leave the system?
I am trying to apply queuing theory / birth and death process to the following.
Suppose customers arrive in a restaurant according to a Poisson process with rate $\lambda = 1$.
Suppose there are $2$ ...
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To what family of densities does $e^{-u}(u^k-k!) \log u$ belong?
Apparently $\int_0^{\infty} e^{-u} (u-1) \log u du = 1$, $\int_0^{\infty} e^{-u} \frac{1}{3}(u^2-2) \log u du = 1$ etc.
Does these densities belong to a known family of densities? The closest I found ...
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When does $k$-th central moment relate to $k$-th power of expectation?
Given a bounded random variable $X$. When exactly is the $k$-th central moment $ E (X-\mu)^k $ upper bounded by the $k$-th power of the expectation $ \mu^k$? This seems to be the case for bounded, ...
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3
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182
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Yule process intuitive question
I have a Yule process with $n$ individuals.
There is no death, so the death rate is $\mu_n$ $=$ $0$ for all $n$.
Each individual gives birth to a new individual independently after waiting for $\text{...
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Maximum of Sub-Exponential and its Tail Probability
Consider zero-mean sub-exponential random variables $\{X_1,...,X_n\}$ (not necessarily independent) with parameters $(\nu, \alpha)$. That is,
$$\mathbb E[\exp\{\lambda X\}] \le \exp\{\nu^2\lambda^2/2\...
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Expressing the Normal Distribution as a Member of the Exponential Family
Background Info:
I am looking for a more general form for the exponential family of distributions. Let's start by considering the normal distribution:
$$p(y|\mu,\sigma^2)= \frac{1}{\sqrt{2\pi} \sigma}\...
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Joint Distribution Function of a Joint Density Function (exponential)
I have a density function $f(x,y)= 2e^{-x-y}$ for $0<x<y<\infty$ and $0$ elsewhere. What is the joint distribution function?
So far I have calculated $F(x,y)= \int_0^x \int_0^yf(x',y') dy'dx' ...
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Finding rates by setting up a birth and death process
I have the following scenario, where I am trying to set up a birth and death process.
There are $10$ bulbs, and the bulbs have independent Exponential $(\lambda)$ lifetimes. If a bulb stops working, ...
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Find the UMVUE of $P(X \leq c)$ in exponential distribution?
Given iid observations $X_1,...,X_n$ on X with the pdf $f_\theta (x)=e^{-(x-\theta)} I(x > \theta)$, find the UMVUE of $P(X \leq c)$, for fixed $c>0$.
My attempt:
I find $P(X \leq c)= 1-e^{-(c-\...
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2
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How to calculate variance in a Poisson process with exponential lifetimes of arrivals?
I am trying to understand how to combine the concepts of Poisson process and the birth and death process.
I have a Poisson process where people arrive with rate $\lambda$ -- so when an event occurs, ...
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How to calculate expected value when there's a race between two exponential random variables?
I have a set of practice questions to prepare for finals. And I have worked through most.
I am unsure about my approach in the below question. I have outlined what I have worked out here.
Suppose ...
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The first holding time of a continuous-time Markov chain is exponentially distributed
Let $X = (X_t)_{t\geq 0}$ be a time-homogeneous Markov process on a finite state space $S$ with right-continuous paths and generator $Q$ (sometimes called Q-matrix). Let $T$ be the first jump time (or ...
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Exponential Waiting times with two queues
I am trying to understand the applications of exponential waiting times when there are two queues.
Let there be two counters in a mall, the first counter $X$ (where the order is placed), and the ...
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During the process of simulating an exponential random variable
According to the text book, to get a associated CDF $$F_x$$ is given by
$$F_x(a)=\int_{-\infty}^af_x(x)dx=1_{[0, +\infty)}(a)\int_0^a\lambda e^{-\lambda x}dx=1_{[0, +\infty)}(a)(1-e^{-\lambda a})$$
...
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exponential distribution for a traffic light [duplicate]
I'm new to this topic and I'm sorry if my question in basic.
Imagine a traffic light that starts working from 7 a.m. every morning, with the frequency of 1 minute green light and 3 minutes red light (...
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Buses arrive at the rate of 9 per hour. What is the probability that you would need to wait for more than 20 minutes if one bus just departed?
Firstly, I understand that this question may be simply carried using the Poisson distribution. However, I attempted this question using the Exponential distribution, but am not getting the correct ...
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Not following derivation of Curie-Weiss-Potts model
I'm reading an article that derives an expression related to the Curie-Weiss-Potts models. The question pertains to how Equation (7) in the article is derived. Below is my summary of the information ...
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Finding variance of random variables that are exponentially distributed
Suppose that lifetime of bulb of type n is exponentially distributed with mean value n years, where $n = 1, 2, 3, 4, 5$.
A bulb is equally likely be chosen from these five types.
(a) Find the ...
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Inverting a moment generating function with simulation
I need to solve the following equation for $\lambda$ involving the moment generating function of a positive random variable, $T$:
$$E_T[\exp(-\lambda T)] = q$$
Here, $0<q<1$ and I am able to ...
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Poisson process with exponential translations
Let $N(t)$ be a standard Poisson process with intensity $\lambda>0$. As we know, for $n \geq 1$ we can define the time of $n$th event as
$
S_n = \sum_{k=1}^n T_k, \; S_0 = 0,
$
where $(T_k)$ are ...
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PDF for total duration of "on" state in a Poisson process with binary values
I have a process where, at times given by a Poisson law of rate $\lambda$, a system picks two values - say "on" or "off" - with respective probabilities $p_0$ and $1 - p_0$. What ...
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69
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Conditional probability with identically distributed variables
I am solving a question using a different approach to the one I have found in my maths material. My teacher presented one solution and mentioned that there several other ways to solve this problem. I ...
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Testing the memoryless property versus independence
I have following sequence of 0-1 values where 1 represents arrival of something, and each 0 and 1 are measured in equal time unit (e.g. every hour). Below is an example of such sequence
...
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Sum of Gamma distribution with different scale.
Let, $X_i$~$exp(\lambda_1)$ and $Y_i$~$exp(\lambda_2)$ iid for i = 1, 2, 3, ....
Define the r.v, $Z_1^k = \sum_{i=1}^{k}(X_i + Y_i)$ and $Z_2^k = \sum_{i=1}^{k}(X_i + Y_i) + X_{k+1}$ for k=1, 2, 3, ....
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Bias of a Maximum Likelihood Estimator
Question:
For $\lambda \subset \mathbb{R}$, define the function
$$
g_\lambda(y)=e^{-(y-\lambda)} 𝟙_{(y\geq\lambda)}
$$
Let $Y=(Y_1, Y_2, \ldots, Y_n)$ with $Y_1, Y_2, \ldots, Y_n$ iid random ...
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PDF of $P(B_1 + B_2 > t)$ where $B \sim \mathrm{expo}(\lambda)$
I am trying to dervie the PDF of $P(B_1 + B_2 > t)$ where $B \sim \mathrm{expo}(\lambda)$.i.e. the probability that the waiting time of two sucessive draws from a exponential distribution will be ...
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Convolution of two exponential functions where x > 0
Assuming $X$ and $Y$ are i.i.d. random variables let $Z = X + Y$
$$
f_X(x) =
\begin{cases}
\lambda e^{- \lambda x} & x \gt 0 \\
0 & \text{else}
\end{cases}
$$
$$
f_Y(y) =
\...
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What is the expectation of switch activation?
Consider $n$ switches, which activate at rate $a_i$, $1\leq i\leq n$. The time it takes for a specific switch to activate, $T_i$, satisfies $T_i\sim \text{Exp}(a_i)$. In particular, $E[T_i]=1/a_i$.
...
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Does the Normal distribution approach an exponential distribution towards $\pm \infty$?
Given the formula for the standard normal distribution:
$$
\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}
$$
There's clearly an exponential in there. Both the normal and the exponential distributions seem ...
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Computing pdf of Y=exp(aX) if X is exponential with mean 1
I'm trying to find the probability $\mathbf{P}(Y<\infty)$ when $Y= e^{aX}$ and $X$ is an exponential random variable with mean 1. Obviously, this will depend on the sign of $a$. If $a=0$ then $\...
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Find the distribution of the sum of ordered exponential elements
Denote a symmetric matrix A =
$$\begin{pmatrix}
a_{11} & a_{12}& ... & a_{1n} \\
a_{12} & a_{22} & ... & a_{2n} \\
...\\
a_{1n} & a_{2n} & ... & a_{nn}
\end{pmatrix}...
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Determine the seller's payment expectations for the warranty [closed]
Suppose the value of an instrument (v) is based on the number of years since purchase (t), thus
$v(t)=e^{7-0.2t}$
If the tool is damaged in the first 7 years since the tool was purchased, then the ...
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Modeling Fog Formation Percentage with Exponential Decay Based on Dew Point Depression and Wind Speed
I am working with a 15-minute time series dataset that includes columns for Dew Point Depression and Wind Speed. Utilizing specific conditions based on these columns, I've successfully categorized fog ...
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Estimation of exponential distribution parameter from smallest $n$ out of N observations
I am interested in estimating the parameter $\lambda$ of an exponential distribution based on the smallest $n$ out of a total of $N$ observations.
In mathematical terms: let $X$ be distributed ...
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Memoryless property of exponentially distributed waiting times in two queues
You are leaving a store, and there are 2 queues with one customer in each. If the waiting times are all exponential random variables, what is the probability that we will be the last to leave?
My ...
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Find density function of sum of exponentially distributed $X_1,X_2$
Let's assume that $X_1,X_2$ are independent and both obey the same exponential distribution
\begin{align*}
&\rho(x)=\begin{cases}
\lambda e^{-\lambda x},& \text{ if } x\geq 0\\ 0, & \text{ ...
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Confusion regarding multiple observers for a memoryless distribution
I have been trying to wrap my head around the memorylessness property of the exponential distribution, and I can't understand the intuition or logic behind it.
Say we have a store where the time ...
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Probability related to compound Poisson distribution
Let $\{Y_n\}_{n\ge 1}$ be i.i.d. exponential random variables with parameter $\lambda>0,$ and let $N$ be an independent Poisson random variable with parameter $\mu$.
Define $X=\sum_{i=1}^N Y_i$.
We ...
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Do exponentially distributed holding times imply Markovian?
I'm trying to understand the significance of the memoryless properties for Markov processes and for the exponential distribution. Both this answer and this answer have been very helpful. However, I am ...
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Find the probability that a second event occurs after time n given than the first event occurs before time n
For the above question, I can solve it quite easily by utilising a counting process however I'm curious as to how one would go solving it by utilising the exponential distribution.
Namely, I'm stuck ...
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Help with law of total expectation and exponential distribution
I have the following question:
A man walks into a bank with one employee. After $T$~$Exp(\mu = \frac{1}{6})$ time another man walks into the bank. If the employee finished helping the first man he ...
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1
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Response times of simulated M/M/1 queue are not exponentially distributed
I have created a simulation for an M/M/1 queuing system in Python using Simpy.
Simulations
The code of the simulation is just one class triggering two processes (in the Simpy sense of process):
One ...
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Equalities involving Exponential Distributions and Markov process
I am trying to understand Remark 17.26 of the book Probability Theory by A. Klenke (3rd version), where the author is showing how a condition on the Q-matrix of a discrete Markov process in continuous ...
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2
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Distribution of the time for the first ocurrence in a Poisson Process given the number of events
I am trying to solve this problem:
Let N(t) a Poisson Process of rate $\lambda$ where ocurrences are type I with probability $p$. Given that $N(t_0) = n$, what is the distribution of the waiting time ...
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Find parameter of exponential distribution (first arrival at $x$ and second at $y$)
I came across this interview question online:
Suppose you sit on the road side and observe cars driving by. Assume the distribution of cars driving by is according to an exponential distribution. Now ...
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Is a sufficient statistic for a parameter $\lambda$ in the exponential family also sufficient for the reciprocal of $\lambda$?
If $\{E(\lambda)\}_{\lambda > 0}$ denotes the exponential family, then $E(\frac{1}{\theta}), \theta > 0$ is an alternative parameterization. I think a sufficient statistic for $\lambda$ is also ...
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derivation of exponential distribution's expectation using poisson
I know that for a Pois$(\lambda)$ distribution, the expectation is $\lambda$. I also know its relation to the exponential distribution, and how you can derive the PMF of the exponential through the ...
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Distribution of minimum of infinite independent and uniformly distributed random variables
We define an infinite sequence of independent random variables $U1, U2, U3, ...$, uniformly distributed across the open interval $(0,1)$.
We also define another random variable, $N=\min \{ n\in \...
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