Questions tagged [exponential-distribution]

A distribution describing the time between events in a Poisson process; a continuous analogue of the geometric distribution.

Filter by
Sorted by
Tagged with
1
vote
1answer
25 views

Find lambda for exponential distribution

I'm trying to find the function which allows me to find lambda if: $$y = 1-e^{-\lambda x}\,.$$ I tried doing this by: \begin{eqnarray} y-1 &=& -e^{-\lambda x}\\ e^{-\lambda x} &=& -...
4
votes
1answer
342 views

Probability of a 500 year flood occuring in the next 100 years - comparison of approaches

I'm looking at this problem A $500$-year flood is one that occurs once in every $500$ years. a) What is the probability of having at least $3$ such floods in $500$ years? b) What is the ...
1
vote
0answers
48 views

Suppose $\frac{1}{Y} \sim Exponential(\lambda)$, what is the pdf of Y? [on hold]

I am actually attempting to answer the following: Suppose $\Pr(X|Y=y) \sim \Gamma(a,y)$ and that $\frac{1}{Y} \sim Exponential(\lambda)$ What is $\mathbb{E}(X)$? However, I know that this is a ...
2
votes
0answers
40 views

Methodology to distinguish single and double exponential distribution

I have a set of n data points $x = [x_1, x_2,...,x_n]$ that follow an exponential distribution or a double exponential distribution (two different $\lambda$ parameters). In order to check which ...
1
vote
2answers
71 views

How does an observation condition the next one, if the numbers are exp. distributued with uknown average?

We have a process that generates exponentially distributed random numbers, i.e., $P(X=x) = \lambda e^{-\lambda x}$. However, we don't know the value of $\lambda$. We observe the first realization with ...
1
vote
0answers
30 views

Testing for Constant Hazard Function

My colleagues and I have some unemployment data, and we'd like to test the hypothesis that it's drawn from a memoryless distribution (= constant hazard function). Is there a standard test for this? (...
1
vote
1answer
59 views

How to change exponential distribution into Normal distribution? [closed]

We have random data, which is exponentially distributed. Data = exp($\lambda$), where $\lambda$ = 0.5. If it is possible to change exponential distribution into the normal distribution. Then what ...
1
vote
0answers
13 views

Can the interarrival times of a continuous time markov chain be distributed with 2 parameter (scale,location) exponential distributions?

I'm trying to model data with a time-homogenous CTMC with a number of states with corresponding constant transition rates $\lambda_{i}$ when I notice that much of the transition times from one state ...
0
votes
0answers
34 views

Product of normal random variables and exponential random variables is exponential random variables?

How we can prove analytically, the Product of normal random variables and exponential random variables is exponential random variables?
6
votes
1answer
166 views

What is distribution of $\sin(x)$? If x is exponential distribution

I am trying to find analytically the distribution of $\sin(x)$, If x belongs to an exponential distribution,
0
votes
1answer
37 views

Probability of 100,000 Computer parts, if one computer part lasts more than seven years is $0.4966$

If the length of time the computer part lasts is exponentially distributed with mean value is $10$. So, for the exponential distribution, we can find the probability of one computer parts. $$p(x>...
1
vote
1answer
67 views

Show that $nX_{(1)}$ is not consistent

Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
3
votes
2answers
84 views

Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
2
votes
2answers
125 views

Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
1
vote
0answers
11 views

How to compare two statistical distributions with unavailability measurements?

I have two different measuring instruments to evaulate if an electronic device is working or not. These instruments provide a working/not-working reading each day and at the end of the month I compute ...
2
votes
2answers
78 views

Hypothesis test for composite null hypothesis of exponential parameter

I'm having trouble defining the reject region based on the generalized likelihood ratio test. This is from a question of past exam I'm self-studying and still have the doubt, given that I got it wrong....
1
vote
1answer
37 views

Re-scaling of exponentially distributed random numbers

I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$ (for simplicity, $N=1$). I found that the generated numbers are initially exponentially ...
0
votes
0answers
21 views

How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
0
votes
0answers
19 views

Why knowing information about something doesn't help when it comes to Exponential Distribution?

I've recently learned about Exponential Distribution and when it's appropriate to be used. There was the following example given. The duration of IC555 integrated circuits under normal operation ...
1
vote
1answer
58 views

Mann-Whitney-U for exponential distributions?

I have two distributions, that look like the following: They appear to be exponential and have different sample sizes (369 vs 60). I would like to do some hypothesis testing. I know that I can use ...
0
votes
0answers
18 views

Questions regarding this derivation of the Poisson Distribution from exponential densities

On page 217 - 218 of the pdf of this book, the author derives the Poisson Distribution using gamma and exponential densities. The author defines $S_n$ to be the sum of a sequence of independent ...
0
votes
0answers
20 views

If there are covariant variables in the exponential distribution, how do I draw the QQ plot?

Observation Data X=x[1],... x[n]; covariant variables are eta=eta[1],...,eta[n]. The parameters of exponential distribution are composed of covariant variables(eta) and lambda. So, X[i]<-rexp(eta[i]...
0
votes
0answers
37 views

To what distribution it's similar? Looks like an exponential but it's not

To what distribution it's similar? Looks like an exponential but it's not. It's seems to have a property, that if I zoom it (xlim) then each time it has the same ...
0
votes
2answers
125 views

Can Kernel Density Estimation estimate an Exponential Distribution?

Can Kernel Density Estimation estimate an Exponential Distribution? I tried to performed to make experiments with various kernels like: "gaussian" and "exponential", but performance seems to be very ...
1
vote
2answers
378 views

How do we build a confidence interval for the parameter of the exponential distribution?

EDIT Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\text{Exp}(\theta)$, where $\theta$ is not known. Precisely, $f(x|\theta) = (1/\theta)\exp(-x/\theta)$ Describe ...
1
vote
1answer
86 views

Why can't I write $P(X>5|X>1) = P(X>5)$? [duplicate]

I have a confusion with the memorylessness property of exponential distribution. If exponential distribution is memoryless (i.e. the past has no bearing on its future behavior), why can't I write $P(...
2
votes
1answer
45 views

exponential RV til bus arrives

Suppose that you are waiting at a bus stop. The waiting time until a bus arrives is $T$ where $T$ is an exponentially distributed random variable with parameter $λ$ i.e. $P(T≤t)=1−e^{−λt}, ∀t≥0$. (a) ...
0
votes
1answer
46 views

Geometric distribution described with rate parameter

I don't understand this sentence from this paper (around equation $5$): The function $H(\tau)$ is the hazard function. $H(\tau) = \frac{P_{\text{gap}}(g = \tau)}{\sum_{t=\tau}^{\infty} P_{\...
2
votes
1answer
43 views

What's the relationship between Pareto shape parameter (alpha) and exponential rate parameter (lambda)?

I'm trying to do my undergraduate research on non parametric density estimation for a heavy tailed distribution. For that, I'm with a data set, which I assumed it should be Pareto distributed with ...
1
vote
0answers
16 views

Passion Distribution [closed]

Suppose the number of tsunami in a season follows a Poisson distribution and the average number of tsunami that hit a region is 5 in every tsunami season A tsunami season lasts for 3 months. In the ...
0
votes
0answers
15 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 ...
0
votes
0answers
14 views

Parameter estimation of exponentialy distributed variable with bounded observation

I have data as below. I believe it is exponentialy distributed. But I could not observe the whole data, since I have limited time. The below table shows the monthly frequency. How can I estimate the ...
0
votes
0answers
171 views

Exponential distribution and Poisson process [duplicate]

Could someone please explain to me what is exponential distribution and poisson process mean? How they are different and the relationship between them? [In simple terms]. Thanks
1
vote
0answers
30 views

Exercise about exponential distribution

My textbook has this exercise, in the section regarding exponential distribution: Given an arrival process with $\lambda = 8.0$, what is the probability that an arrival occurs in the first $t = 7$ ...
1
vote
2answers
78 views

Estimator for $\frac{1}{\lambda}$ using $\min_i X_i$ when $X_i$ are i.i.d $\mathsf{Exp}(\lambda)$

Let $X_1,\ldots,X_n$ be i.i.d. $\mathsf{Exp}(\lambda)$ random variables, where $\lambda$ is unknown. Consider $f_{\min}(x) = \min_{i}(X_i)=$ $ n \lambda $ Exp$(n\lambda x)$. I am told that $\hat \...
0
votes
0answers
108 views

How do I calculate a Bayesian Posterior Distribution from an Exponential Prior and Sample Data

I have a dataset where each observation is a length of time (e.g. 50 days, 70 days, 105 days) and I am trying to utilize Bayesian statistics to calculate a posterior distribution in light of new data. ...
1
vote
1answer
75 views

Proof for simulation of NHPP by thinning

Background: I'm trying to show equivalency between the density function for a non-homogenous exponential process (NHEP?), (i.e. the arrival times of events generated by a non-homogenous Poisson ...
2
votes
0answers
25 views

Trouble understanding derivation of probability for continuous time markov chain

I'm working on exercise 6.10 from "Introduction to probability models" by Sheldon M. Ross. There's an expression for the probability $P_{00}(t)$ that I don't understand. Here's the relevant ...
0
votes
1answer
212 views

Minimal sufficient statistic for location exponential family

Let $X_1,\dots,X_n$ iid with pdf $$f(x|\theta)=e^{-(x-\theta)},\,\,\,\theta<x<\infty,\,\,\,-\infty<\theta<\infty.$$ Part (b) of Problem 6.9 in Casella and Berger asks to find a minimal ...
0
votes
1answer
164 views

Expectation of standard exponential squared given sum of two standard exponentials

So I have been working on this question for a while and made some progress , but I run into a problem about the normalizing constant. The question is, for $X$ and $Y$ i.i.d. standard exponential, find ...
0
votes
0answers
21 views

Explanation for Cumulative Distributive Function example

I'd like to ask for clarification of the following example in my textbook. Example: Suppose events are occurring at random with average rate $\lambda$ per unit of time. What is the probability ...
0
votes
0answers
66 views

Confidence interval with only one observation

How could one create a 95% lower confidence interval for the expectation of a exponentially distributed r.v. with only one observation of the r.v., say 5555?
2
votes
1answer
266 views

CDF and MGF of a Sum of a discrete and continuous random variable

I am currently dealing with the following exercise: Given the random variables $X \sim Be(p), Y \sim Exp(\lambda)$, and assume they are independent. Set $Z:= X + Y$. Compute the Moment Generating ...
1
vote
1answer
337 views

The distribution of the product of a Bernoulli & an exponential random variable

Let $X$ be an exponential random variable $f(x) = c e^{-c x} \text{ if }x > 0; 0 \text{ otherwise.}$ Let $Z$ be a Bernoulli RV with $Pr(Z=1)=0.45$ and $Pr(Z=0)=0.55$. $X$ and $Z$ are independent. ...
3
votes
2answers
213 views

Prove convergence in distribution for n times the minimum of an unknown positive distribution

Let $Z_1, Z_2, ...$ be independent and identically distributed random variables with some density $f$. Suppose that $P(Z_i > 0) = 1$, and that $$ \lambda = \lim_{x\to 0} f(x) > 0$$ Let $X_n = ...
2
votes
2answers
203 views

Regression model when the dependent and independent variables show exponential distribution

As the Title suggests i am trying to figure out what would be the regression model to use when both the dependent and independent variables show an exponential distribution. Do I have to perform a ...
1
vote
0answers
120 views

Can you have an exponential distribution where x is negative? [closed]

I have a random variable with an exponential distribution and have solved an inequality to determine the maximum a posteriori rule (where if $x > \alpha$, I will choose hypothesis 1 over hypothesis ...
0
votes
1answer
188 views

How could “sum of exponential distribution is 1” be proven?

$$f(x; \lambda) = \begin{cases} \lambda e^{-\lambda x} \quad \text { for } x \geq 0 \\ 0 \quad \quad \quad \text { for }x < 0\end{cases} $$ How can I prove that the sum of probabilities under ...
3
votes
0answers
153 views

How do we show the exponential distribution has maximal entropy on R+?

I’ve been looking around the internet, and having trouble finding a demonstration that the exponential distribution is maximal entropy on R+. I’d appreciate any points in the right direction.
8
votes
1answer
2k views

Distribution of sum of exponentials

Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$. Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-...