Questions tagged [exponential-distribution]

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

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Suppose that X and Y are independent exponential random variables each with mean 1

Suppose that $X$ and $Y$ are independent exponential random variables each with mean $1.$ What is $P\!\left(Y > X^2\right)?$
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Confidence interval for the maximum likelihood estimate of the minimum of a left truncated exponential distribution

I am currently working on a problem in which I have observations $y_{i}$ that are distributed, $y_{i} \sim \textrm{Exponential}(\beta = ax_{i})\cdot T[b, \infty)$ where, $\beta$ is the rate parameter ...
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36 views

conditional distribution in coin tossing problem

Let $X$ and $Y$ be exponential random variables with parameters 1 and 2. A coin has probability of getting heads as $p$ and probability of getting tails as $1-p$. Let $Z$ be another random variable ...
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31 views

What is the distribution of gap lengths in a Poisson process?

In a Poisson process with a finite period (and a known long-term-average event rate), what is the distribution of gap lengths between events? The number of events within a fixed period will be given ...
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Significance level of a continuous distribution

Can the significance level of a continuous distribution be exactly equal to its value? For example for Exponential Distribution can the significance level of a test be exactly 0.02?
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Non-linear regression of exponentially distributed data?

I have a family of random variables $y(x)$ and a set of data points $(x_i, y_i)$. I know that, for each $x$, $y(x)$ is exponentially distributed. I have a hypothesis that the mean of the distribution ...
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38 views

Minimizing Mean Square Error

Suppose we have a random sample $\textbf{X}=(X_1,...,X_n)$ from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & ...
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1answer
28 views

Flat “geometric distribution” by varying the probability of the Bernoulli trail

In a simulation I am working on, each day (time step) there is a chance that a condition changes (at which point it is stuck in the changed condition). Setting this probability to a fixed value (say 5%...
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generating a uniform random variable from the first digit of an exponential random variable?

in "introduction to probability models", Ross talks about simulating with the rejection method, and he needs an exponential random variable, and a uniform random variable (used only for checking ...
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33 views

Conditional and joint distribution of the sum of exponential RVs

Let $X_1,X_2,...,X_n$ be i.i.d. $Exp(\lambda)$ random variables and $Y_k =\sum^{k}_{i=1}X_i$, $k = 1,2,...,n$. a) Find the joint PDF of $Y_1,...,Y_n$. b) Find the conditional PDF of $Y_k$ ...
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Sensitivity analysis with an exponential distribution

Statistics newbie here: If you had to run a local sensitivity analysis for an exponential distribution, would you rather change the λ (e.g. with a 5% increment) or would you take your initial value (...
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1answer
37 views

Finding the pdf of $X_{(1)}$ of the two-parameter exponential distribution

I have to find the pdf of the smallest order statistic $X_{(1)}$ of two-parameter exponential distribution whose pdf is: $f(x; \theta_1, \theta_2) = \frac{1}{\theta_2} \exp\{-\frac{x-\theta_1}{\...
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49 views

Transformation of a random variable with a gamma distribution

Suppose $X_i \stackrel{i.i.d}{\sim}$ Exp$(1/\theta)$ which implies $\sum_{i =1}^{n} X_i \sim$ Gamma $(n, 1/\theta)$. But, then, the book that I am reading says that $(2/\theta)\sum_{i =1}^{n} X_i \...
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Local sensitivity analysis with exponential and uniform distributions as input

For my thesis I need to run a sensitivity analysis on the input factors for a supply chain model. I am supposed to change the mean and the standard deviation (sd) of all input factors respectively by ...
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1answer
39 views

Estimating mean values given a histogram (without using median point)

I have a list of stock trades made by US senators and the exact amount of money they spend on a trade is not disclosed, they only give a range of values. Below is the frequency table for the 8600 ...
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1answer
46 views

Variance of $\frac{X_i}{\theta^2} -\frac{1}{\theta}$ in an exponential distribution

I read in a book discussing the exponential distribution that the variance of $\frac{X_i}{\theta^2} -\frac{1}{\theta}$ is equal to $\frac{1}{\theta^4}Var(X_i) = \frac{1}{\theta^4}\theta^2$. Can ...
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What is the distribution of a mixture of exponential distributions whose rate parameters follow a gamma distribution?

I want to know the theoretical distribution of a mixture of exponential distributions whose rate parameters are distributed according to a gamma distribution: $$ y\sim\text{Exp}(\theta), \quad\text{...
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Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...
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Maximums of two exponentials

The wikipedia page for the exponential distribution states that for $X_1, X_2, \dots X_n$ independent exponentially distributed with rate parameters $\lambda_1,\lambda_2,\dots,\lambda_n$, the index of ...
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1answer
60 views

Can I use z-scores with an exponential distribution? Or is there another test statistic for these types of distributions?

I have an exponential distribution for a population. $\theta = \mu = \sigma$ is known. Sample size $n$ is known. I need to find $"𝑃(𝑎<𝑋¯<𝑏)"$ for a random sample with size $n$. I think I am ...
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1answer
29 views

Sample Size for Exponential Distribution

I'm having a bit of trouble with a simple question. I have data for a product, amount and days on which it was sold, that when plotted as a cumulative histogram follows an exponential cdf. I want to ...
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1answer
148 views

Sum of Exponential and Gamma Distributions [duplicate]

I have been learning sums of distributions and understand that the sum of exponential distributions with parameter B is a gamma distribution with parameters a=1 and B. However, I need to figure out: ...
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259 views

How could one (tentatively) estimate the actual number of COVID-19 infections in an area, using hard data like age-adjusted death rate?

A big problem with the current COVID-19 epidemic is the difficulty of getting tested (due to mild symptoms and lack of testing kits). This makes it impossible for patients with little or no symptoms ...
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1answer
71 views

Linear Model, Distribution of Maximum Likelihood Estimator

Let $\epsilon_i \sim \text{Exp}(\lambda)$, $\lambda > 0$, and iid for all $i = 1,2, \dots$ Suppose for we have the linear model $$ Y_i = \beta X_i + \epsilon_i,$$ where $X_i > 0$ for all $i =...
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Distribution of the median of an exponential distribution?

Say i have $X_{ij} \sim Exp(\theta)$ for $i=1...n,\;\;j=1...k$. Afterwards I derive $k$ medians for $k$ groups of size $n$ (even number). The median is a random variable say $M_j$. Do we have any ...
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Quantile-calculation example: Why the term $c$?

I need help to interpret a solution to the following example: Based on historical claim amounts $x_1,...,x_{47}$, which are assumed to be outcomes from an unknown claim distribution, you are ...
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3answers
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Fitting data to an exponential model with a specified rate in R

I have been using the fitdistr package in R to try and do this but with no luck so far: fit1 <- fitdistr(data1$x, "exponential", start = list(rate = 10)) I am ...
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1answer
30 views

Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?

Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x}, \;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda e^{-\lambda x},\;\; y>0.$ Find the probability ...
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105 views

Is there a model to predict mortality rate of an ongoing epidemic?

Following the recent Coronavirus outbreak, various mortality rates have been published. Most of these are simply the ratio between the deaths and the total verified cases, which is not a very ...
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Maximum Likelihood Estimator with exponential noise

So I need a little help with this please. I'm given N measurements of a signal $Y_{i} = A + v_{i}, i = 1,...,N$, where $v_{i}$ is measurement noise with the exponential pdf $f_{v}(v) = e^{-v}, v \geq ...
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1answer
42 views

Estimating a distribution from sums of samples

I'm trying to figure out the parameters of a distribution from real data, but I only get their sums and counts. For either exponential or normal distributions. So, I'll get the sum of 27 samples, ...
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What is the PDF for the minimum difference between a random number and a set of random numbers

I have a list (lets call it $ \{L_N\} $) of N random numbers $R\in(0,1)$ (chosen from a uniform distribution). Next, I roll another random number from the same distribution (let's call this number "b")...
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Q-Exponential distribution in Numpy

I am studying a phenomena, which I know is characterised by the q-exponential distribution, with parameters $c=1$, $q = 1.355$ and $b = 0.524$ in the equation $$e_{q,b,c}(t) = c(1 + b(q-1)t)^{\frac{1}{...
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1answer
48 views

Exponential distribution with mean 1/lambda [duplicate]

Let x1,x2,x3,.......xn be independent and identically distributed random variables from exponential distribution with mean 1/Lambda. Let Sn=x1+x2+......+xn and N=infimum(n>=1:Sn>1) then what is the ...
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1answer
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MLE $\hat{h(\mu)} = h(\hat{\mu})$ of $h(\mu) = var(Y_1) = \mu^2$

Question: Suppose Y1, · · · , Yn follows an Exponential distribution with $\lambda = \frac{1}{\mu}$. Derive the MLE $\hat{h(\mu)} = h(\hat{µ})$ of $h(µ) = var(Y_1) = µ^2$, and show that $h(\mu)$ is ...
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Convolutional roots of mixture of exponential distributions

In the reference book on infinite divisibility and generalised gamma convolution, BONDESSON, Lennart. Generalized gamma convolutions and related classes of distributions and densities. Springer ...
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100 views

How do I calculate Confidence Interval for Gamma Distributed Pivotal Quantity?

I'm studying confidence intervals and then I came across the following problem: It's said that a random variable X has Skewed Exponencial Distribution with parameters $\alpha >0$ and $v \in \...
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problem in understand exponential PDF [duplicate]

I'm studying a paper called "Optimization based on bacterial chemotaxis". As it can be understood from its name, it has proposed an optimization algorithm based on the reaction of a bacterium toward ...
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1answer
94 views

Question about an exponential Bernoulli distribution

Mohie El-Din and Amein (2011) define a distribution in formula (1.2) which they call the exponential Bernoulli distribution (EBD). The distribution has the following form: $$\displaystyle f \left(t \...
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a question about method of moment estimator

I have a question about method of moment estimator. Say I have a IID sample $X_1, X_2, ..., X_n$ from an exponential distribution $Exp(\theta$), say I want to find the method of moment estimator of $...
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Can we consider arrival of pest outbreaks reports as Poisson Process?

I have a scenario where given a geographical region, farmers within that region can generate pest outbreak reports (like twitter messages) using a mobile phone. Also, a particular pest can spread to ...
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1answer
104 views

inverse of an exponential distribution

I have a question regarding this. Say I have $X_1, ..., X_n$ be random sample from an exponential distribution i.e. $Exp(\theta)$, and let $\gamma = \theta^2$. Let denote $\gamma^{mme}$ as the ...
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1answer
114 views

Bivariate exponential distribution $(S, T)$ with controllable correlation and $S\leq T$

I am trying to define a bivariate exponential distribution $(S, T)$ with marginals $S\sim\mathrm{Exp}(\lambda_S)$ and $T\sim\mathrm{Exp}(\lambda_T)$ for $\lambda_S > \lambda_T$. I would like the ...
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2answers
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Finding the probability of survival of an insurance company

I was given as a homework exercise the following problem: however, I came into a disagreement with one of my classmates. Given that the solution is not shown, I was wondering whether mine was correct....
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1answer
51 views

Hazard rate with exponential distribution

Hi I was trying to understand hazard rate and got stuck in the middle. Any suggestions are welcome. Below is the problem. Consider the following distribution for the duration of an unemployment spell ...
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39 views

Birth and death process, and calculating waiting time using Little's law

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible ...
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1answer
34 views

Finding limiting probability for continuous-time Markov chain

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible ...
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0answers
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Incorporate sales forecast uncertainty into product stock availability simulation

I work in a e-commerce and I’m estimating availability of our catalog products. The process (in its simplified form) looks the following. A customer order comes in and we fulfill it if we have stock ...
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1answer
123 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} &=& -...
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1answer
388 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 probability ...