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Questions tagged [exponential-distribution]

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

13
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4answers
23k views

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 ...
12
votes
3answers
2k views

What's the kurtosis of exponential distribution?

Original question (with confused terms): Wikipedia and Wolfram Math World claim that the kurtosis of exponential distribution is equal to $6$. Whenever I calculate the kurtosis in math software (or ...
8
votes
2answers
3k views

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}...
7
votes
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 ...
6
votes
2answers
15k views

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} \...
6
votes
1answer
386 views

Maximum of exponentials divided by sum

Let $X_1,\dots, X_n$ be i.i.d. exponential random variables. For large $n$, what is the probability distribution for the following? $$\frac{\max X_n}{\sum X_n}$$ I believe that the cdf of the ...
6
votes
0answers
134 views

Poisson Process: indepedent increment

Let $\{N(t): t\geq0\}$ be a Poisson process of rate $\lambda$, and let $S_n$ denote the time until the $n_{th}$ event occurs. compute $P(S_3>5|N(2)=1)$ Attempt: Notice that $P(S_3>5)=P(N(5)&...
6
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0answers
142 views

Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
5
votes
2answers
593 views

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: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
5
votes
3answers
975 views

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 ...
5
votes
2answers
431 views

Random variable with exponential distribution.

Let $X$ be random variable with exponential distribution $\mathcal{E}(2)$ and let $Y$ be another random variable such that $$Y=\max\left(X^2, \frac{X+1}{2} \right).$$ Find the distribution for random ...
5
votes
2answers
68 views

Fundamental Identities on Exponents

I came across some properties of "Exponential and Logarithmic Equations and Inequalities" in the book "Problems in Mathematics" by V GOVOROV. The problem which I am facing is of one of the properties ...
5
votes
0answers
117 views

First moments of Geometric Brownian Motion-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^...
5
votes
0answers
60 views

Find probability density function for $\varepsilon \cdot X$.

Let $ X $ be exponentially distributed with parameter $\lambda > 0$ . $a)$ Find the probability density function for $ Y := exp(X) $. $b)$ Consider $X$ again. Now let $\varepsilon$ be an ...
4
votes
2answers
2k views

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 ...
4
votes
2answers
7k views

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 ...
4
votes
3answers
569 views

Making Sense of the Exponential Distribution and the Probability Density Function

I read that, due to the memoryless property of exponential distributions, the distribution should be used when the rate of an event is constant during the entire period of time. An example would be ...
4
votes
1answer
141 views

Finding the distribution of the reciprocal of a random variable

Let $X\sim \text{Exp}(\lambda)$ be an exponentially distributed random variable. That is it has the probability density function $f(x)=\lambda e^{-\lambda x}1_{[0,\infty)}(x)$ and cumulative ...
4
votes
1answer
789 views

Waiting time: exponential distribution

Smith is waiting for his two friends Lee and Yang to visit his house. The time until Lee arrives is Exp($\lambda_1$) and the time until Yang arrives is Exp($\lambda_2$). After arrival, Lee stays an ...
3
votes
2answers
182 views

If the expected value of $X^n$ is $n!$, what is the probability density function of the random variable $X$?

I am working through my homework and this problem has me stumped. I don't know how to come up with a pdf just by being given an expected value? We are learning about exponential distributions, but ...
3
votes
2answers
112 views

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 $ \...
3
votes
1answer
170 views

In the limit of $N \rightarrow \infty$, find solution $z$ to $\text{e}^{-(z+N)} \sum \limits_{k=0}^{N} \frac{(z+N)^k}{k!}=\frac{1}{2}$

Fix an integer $N$, and consider the unique positive solution $z$ to the following equation: $$\text{e}^{-(z+N)} \sum \limits_{k=0}^{N} \frac{(z+N)^k}{k!}=\frac{1}{2}$$ For $N = 0$, we find that $z ...
3
votes
1answer
21 views

Cumulative Distribution Function of a Variable with Exponential Distribution

Suppose we have a variable $X$ that has an exponential distribution with a probability density function: $f(x) = 3e^{-3x}, x >0$ Then the cumulative distribution function is: $\int_{-\infty}^...
3
votes
2answers
2k views

What do we mean by rate in the exponential distribution?

When we talk about $X$ being a RV with an exponential distribution $$f(x)= 1-e^{-\theta x} \,\,\,\,\, \text{for}\,\,\,\ \theta>0$$ we say that it describes the time between two events in a Poisson ...
3
votes
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. ...
3
votes
3answers
150 views

Suppose that $X$ has the exponential distribution. Find the density for $X^3$

Suppose that $X$ has the exponential distribution. Find the density for $X^3$. Really not sure where to go with this problem, my notes from class weren't sufficient and after poking around online ...
3
votes
1answer
64 views

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 ...
3
votes
2answers
279 views

Help with intuition on the exponential distribution

My book says an exponential random variable is good for modelling the waiting time until the ocurrence of an event. However, I'm having a bit of difficulty understanding why that should be the case, ...
3
votes
2answers
75 views

Exponential conditional probability

There are two types of claims that are made to an insurance company. Let $N_i(t)$ denote the number of type $i$ claims made by time $t$, and suppose that $\{N_1(t), t \ge 0\}$ and $\{N_2(t), t \ge 0\}$...
3
votes
1answer
244 views

Covariance between an exponential random variable and the maximum of several exponential random variables

Suppose $X_1, \ldots, X_n$ are i.i.d. exponential RV with parameter $\lambda$. Let $Z = \max\{X_1, \ldots, X_6\}$. My goal is to find $\mathrm{cov}(X_1, Z)$. I already know the c.d.f., p.d.f., and ...
3
votes
1answer
175 views

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 ...
3
votes
1answer
206 views

Probability that one station becomes empty before another.

Question: There are 2 stations A and B in series having i and j customers respectively. Customers after being served at station A are routed to station B. The service time of each of the queues are ...
3
votes
1answer
1k views

Distribution of Square of Rician Random Variable?

We know that the square of a Rayleigh random variable has exponential distribution, i.e., Let the random variable $X$ have Rayleigh distribution with PDF $$f_X(x)=\frac{2x}{\alpha}e^{-x^2/{\alpha}}.$$...
3
votes
1answer
32 views

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 ...
3
votes
1answer
147 views

Conditional waiting time in Poisson process

Consider a homogeneous Poisson process with inter-arrival times $T_i$, which follows the exponential distribution with rate $\lambda$. Let $N(t)$ denote the number of arrivals by time $t$. Suppose I ...
3
votes
1answer
52 views

probability that minimum value in set A is larger than maximum value in set B

There is a two sets, called $A$ and $B$ which have $K$ number of elements, respectively. Each element in both group follows one of two exponential distribution. One is $f_X(x) = \alpha e^{-\alpha ...
3
votes
1answer
109 views

Finding the expected value in a random drawing

Suppose the life span of a battery follows exponential distribution with the expected value of 1 year. If three battery are drawn at random and tested until they all fail, what is the expected life ...
3
votes
1answer
30 views

How many cells will be born before it dies?

A given cell can replicate at rate $\lambda$ and die at rate $\mu$. Upon replication, the cell divides into 2. The question asks: how many cells will be produced by this cell before it dies? My ...
3
votes
1answer
279 views

Relation between Hazard rate and expected time until next failure.

For an exponential distribution, the hazard rate has a clear interpretation as the inverse of the expected time until the next failure (let's say we are modelling machine failures here). For other ...
3
votes
1answer
61 views

customer service time problem

In a disc shop two employees are working. When we get inside the shop, we see that the two employees are already serving two customers (one customer for each employee), with the service time being a ...
3
votes
0answers
1k views

Method of moments exponential distribution

Find the method of moments estimate for $\lambda$ if a random sample of size n is taken from the exponential pdf, $$f_Y(y_i;\lambda)= \lambda e^{-\lambda y} \;, \quad y \ge 0$$ $$E[Y] = \int_{0}^...
3
votes
1answer
214 views

Gamma distribution family and sufficient statistic

Let X $=X_1,...X_n$ be a random sample iid from the probability density function: $$ f(x;\theta)=\frac{\Gamma(\theta)\sin(\pi\theta)\theta^{1-\theta}}{\pi}e^{-\theta x}x^{-\theta}$$ $x>0, 0<\...
3
votes
0answers
1k views

Probability - long-run proportion

So the problem asks: A zoo owns a population of rare Galapagos turtles. The turtles bring in a lot of visitors but unfortunately do not breed in captivity. To maintain the population, new turtles ...
3
votes
1answer
95 views

Stationary distribution - Time of service [closed]

There's in a bank two identical queues and totally separated : these are two queues of type $M/M/1$. For each of them, the arrivals are separated by exponential times of parameter $\nu$, the time ...
3
votes
1answer
103 views

A Nonhomogenous Poisson Process Question - Harry's Stressful Life

Due to stress of coping with business, Harry begins to experience migraine headaches of random severities. Headaches are instantaneous and has zero duration. The times when headaches occur follow a ...
2
votes
2answers
51 views

Expectation when x>5

Exam question was: Suppose $X$ is an exponential random variable with the expectation of $1$, find $E(X|X>5)$. My answer was that first find $$P(X<x | X > 5)$$ Which I got to be $1-e^{5-...
2
votes
2answers
169 views

Awesome riddle including independence and exponential distribution [closed]

The life cycles of 3 devices $A, B$ and $C$ are independent and exponentially distributed with parameters $\alpha,\beta,\gamma$. These three devices form a system that fails if not only device A fails ...
2
votes
4answers
1k views

What is the distribution of 1/($X$+1)?

I have a problem that I'm having trouble figuring out the distribution with given condition. It is given that 1/($X$+1), where $X$ is an exponentially distributed random variable with parameter 1. ...
2
votes
2answers
182 views

Renewal process problem, where $X_i$'s are i.i.d. with exponential distribution.

A room is lit by $2$ bulbs. Bulbs are replaced only when both bulbs burn out. Lifetimes of bulb's are i.i.d exponentially distributed with parameter $λ=1$. What fraction of the time is the room only ...
2
votes
2answers
95 views

Total waiting time of exponential distribution is less than the sum of each waiting time, how so?

I am reading my textbook and find a weird phenomenon. The example says that Anne and Betty enter a beauty parlor simultaneously. Anne to get a manicure and Betty to get a haircut. Suppose the time for ...