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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Write the PDF of an exponential prior given E[$\theta$] = 2

I am reviewing old exercise solutions and the following info is given: Assume that the conjugate prior for θ (as a special case of the gamma distribution) is following the exponential distribution ...
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Find pdf of X+Y [duplicate]

Let X ∼ Exp(λ) and Y ∼ Exp(μ) be two independent exponential random variables, where λ, μ > 0. Find the probability density function of X + Y if λ ̸= μ. I have successfully find ans if λ = μ, but ...
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What Explains the Behavior of these Graphs?

I thought of the following experiment: Suppose we generate a single random number from a normal distribution (with a specific mean and specific standard deviation), we then take the difference of this ...
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What are the operational links between exponential and weibull distribution

I recently read a very interesting article about competing risks that came with simulations. In these, the author defines a Weibull distributed variable w using an ...
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What is the name of the functions in exponential dispersion family?

If an exponential family is given by: $g(y|\theta) = exp\{\theta^TT(y)-A(\theta)\}h(y)$ then the functions $h(y)$, $A(\theta)$ and $T(y)$ are defined by names: $T(y)$ is a sufficient statistic $A(\...
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Calculating expected value

Came across an interesting problem. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it to the ...
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Correlation and Max and Min of Two Exponentials [duplicate]

Came across an interesting problem: Let X and Y be independent random variables such that both X and Y ∼ Exp(1). Define L = min(X,Y) and H = max(X,Y). What is ρ(L, H)? Why must the correlation should ...
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Generating multivariate random variable with normal and exponential marginals

I have a collection of data points of the form $[U, V, X, Y]$, where $U$ ~ $N(\mu_1, \sigma_1)$; $V$ ~ $N(\mu_2, \sigma_2)$; $X$ ~ $exp(\lambda_1)$; and $Y$ ~ $exp(\lambda_2)$, and I am looking to ...
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Unexpected distribution of ab-cd where a,b,c,d are independent and N(0,1) distributed

I have discovered an unexpected curiosity that if a,b,c,d are independent random variables and are N(0,1) distributed, then |ab-cd| is exponentially distributed Exp(1). It's easy to verify that ...
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Solving for the parameter of an exponential distribution

Suppose I have a random variable $X$ where $X$ follows an exponential distribution of the following form: $$f_X(x) = \frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$. I want to find the value of $\lambda$ ...
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Exponential random variable X with a uniform random variable as its parameter

$$X\ \sim Exp(U) ~ and\ U\ \sim U(0,1) $$ The question asked for the value of $ P(X\geqslant 1)$ I saw the solution and it went like this: $$P(X\geqslant 1) = E[P(X\geqslant 1)|U] = E[e^{-u}] = \int_{...
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How to change some parameters of a process to make it WSS?

Consider the R.P., $X(t)=Au(t-T)$ where, $$A \sim N(\mu,\sigma^2)$$ $$T \sim Exp(\lambda)$$ and $u(t)$ is the unit step function. If $A$ and $T$ are independent we'll have, $$E(X(t))=\mu(1-\exp\{-\...
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Some help on this question about stochastic epidemic models without removals would be greatly appreciated

I get this question up until the underlined part, would someone be able to explain the rest of it to me as its had me stumped for a little while now.
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Visualizing relationships in log-link/exponential distribution models by placing the linear predictor on the Y axis?

I'm visualizing results from a negative binomial regression. I don't want to the graph of Y vs X to look exponential, I want it to look linear. In SPSS, the value provided for the linear predictor is ...
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Light Bulb hypothesis testing

One claims that the life time distribution of its Everyday light bulbs is exponential with mean 1000 hours. If you test a random sample of 4 light bulbs and find that the average life time is 900 ...
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Find UMVUE of difference of parameters of two exponential distribution random variables

Let $X_{1}, \dots, X_{n}$ be i.i.d. having the exponential distribution $Exp\left(0, \theta_{x}\right)$ with $\theta_{x}>0$, and $Y_{1}, \dots, Y_{n}$ be i.i.d. having the exponential distribution $...
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Sum of exponential decays with normally distributed lifetimes

If I have the sum $$f(t)=\frac{1}{N}\sum_{i=1}^N e^{-\frac{t}{\tau_i}}$$ where each $\tau_i$ is distributed normally with mean $\tau$, what will be the mean functional form of $f(t)$ in the large $N$ ...
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What is $E[X]$ and $\text{Var}(X)$ if $X$ follows $Pois(T^2)$ and $T$ follows Exponential distribution [duplicate]

I'm new to this community. I have problem in finding expected value and variance of R.V.s that are composed of other R.V.s following other distributions. Suppose $X \sim Pois(T^2)$ where $T \sim Exp(\...
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Why my cdf of the convolution of n exponential distribution is not in the range(0,1)?

I assume that there are n exponential distribution that $x_i$ ~ $Exp(\lambda_i)$, i=1..n, and I want to calculate the cumulative distribution of $S=x_1+x_2+...+x_n$, the convolution of n exponential ...
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Equally spaced points on x-axis in exponential distribution probplot/qqplot

I have a sample distribution and want to check if it is exponential. For that I am trying a qqplot and probplot. ...
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Help understanding gamma posterior of exponential likelihood

The posterior of $\text{Exp}(x;\lambda)$ with prior $\text{Gamma}(\lambda;\alpha, \beta)$ is $\text{Gamma}(\lambda|\alpha+n, \beta + n\bar x)$ where $n$ is the number of observations and $\bar x$ is ...
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Expected value of maximum of $n$ iid exponential random variables [duplicate]

I was recently playing around with this distribution. Let $Y_n \sim \max_i X_i$ where $X_i \sim \exp(\lambda)$. Then the well-known result $$ f_{Y_n}(y) = \lambda n e^{-\lambda y}(1-e^{-\lambda y})^{n-...
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Distribution of a fraction of exponential random variables

Let $X_1, X_2$ be independent exponential random variables with common pdf $f(x)=\lambda\exp(-\lambda x), x>0$. How do I show that $Z=X_1/(X_1 + X_2) \sim U(0,1)$? I know that $F_Z(z) = P(\frac{X_1}...
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Exponential Posteriori with a Uniform Prior

I'm studyng for a final exam and found this problem from another generation, but I don't know how I should continue... I will be gratefull for any help, thanks you. Let be $X|\theta\sim U(0,\theta)$ ...
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How to use the exponential distribution to generate samples from the chi-square and beta distributions?

I am supposed to use rexp() in R to draw from an exponential distribution with mean 1, and then use those draws to generate 1000 draws from each of the following: ...
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Confidence interval for exponentially distributed estimator

We have an estimator $\hat{\theta}\geq 0$ for $\theta$, with distribution function $P\{\hat{\theta}\leq t \}=1-e^{-t/\theta}$, which we can recognize as the cdf of the exponential distribution. Our ...
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How to find an unbiased estimator for reciprocal of scale parameter given an iid exponential sample?

For a random sample $X_1, ..., X_n$ from an exponential distribution with scale parameter $\lambda$, the density is given by $f(x) = \frac{1}{\lambda}e^{-\frac{1}{\lambda}x}; \,x \geq 0,\, \lambda >...
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How to prove that t = min{t1, t2} follows an exponential distribution if t1, t2 follow another different exponential distributions

I have no idea about how to prove the next: Suppose we have two random variables, $t_1$ and $t_2$, that follow the distributions $\lambda_1e^{-\lambda_1 t_1}$ and $\lambda_2e^{-\lambda_2t_2}$, ...
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Is it a Poisson or an Exponential distribution?

I have to do statistical inference (homework) on following data and I am struggling to choose if I should go with an exponential or a poisson distribution to do my analysis. Can you give me any hints? ...
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Exponential distribution confidence interval

Question: I think I mostly understand (a) and (b), but I don't understand (c). My Work: Asymptotically, $2\big[ \ell(\hat \lambda) - \ell(\lambda) \big] \sim \chi_1^2$, where $\hat \lambda = 1/\bar X$...
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Distribution of the exponential of an exponentially distributed random variable?

Let $X$ be an exponentially distributed random variable, that is, with density function $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ ($\lambda>0$), and cdf $F_X(x)=1 - e^{-\lambda x}$. What is the ...
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How to find the right scaling for exponential distribution

I have a complex Gaussian variable, $Z=X+jY$ with $X,Y \sim \mathcal{N}(0,\sigma^2)$, and I would like to find the parameter that scales the distribution of the squared magnitude $P=|Z|^2$. As ...
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Confidence Interval for exponential distribution using pivot quantity

Let' say that $X \sim exp(\theta)$. And we have a sample of size $n$ of $X$ and we consider as an estimator $\hat{\theta} = X_{(1)} = min\{X_1,...,X_n\}$ and also consider $Y = \theta X_{(1)}$. a) ...
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UMVUE for P(X > k) in exponential distribution [duplicate]

I have to find UMVUE for $exp(-k*a)$ where X ~ Exponential(a); k is a positive real number. I tried it using Lehmann-Scheffe theorem. Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
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Modeling decreasing price elasticity of a good

I am attempting to model the decreasing price elasticity/response for a good. I need to control for place and time features and available alternatives. Besides this, I also need to add time and ...
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If LASSO is equivalent to Bayesian Regression with a Laplace (double exponential) prior, what would be the prior for non-negative LASSO? Exponential?

We know that the LASSO penalty is equivalent to Laplace prior. So what would be the corresponding prior for a non-negative LASSO? Is it exponential distribution? More generally, is it true that every ...
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Find the expectation of an exponential distribution estimator

So we've got a sample data coming from exponential distribution with parameter $\lambda$, and we take an estimator $\lambda_n = \frac{n}{X_1+X_2+\cdots+X_n}$. I need to show that this is a biased and ...
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1 vote
1 answer
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Probability of joint dependent events

I'm having trouble finding a way to do this calculation and checking if I'm correct: Let $X_1 \sim Exp(2)$ and $X_2 \sim Exp(2)$ be independent random variables $\left(f_X(x) = 2e^{-2x}\right)$, ...
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Expected Waiting Time for Each Person in a Queue

Suppose I have the following problem: There is a coffee shop with 2 employees (server) taking orders. Customers come to the coffee shop and form a queue, based on "first in, first out". ...
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Insurance Claims: Proving a Process is a Poisson Process and Finding its Rate

Let $X(t)$ denote the number of claims received by an insurance company in the time interval $[0,t]$. We will assume that ${X(t) : t ≥ 0}$ can be modelled as a Poisson process, where $t$ is measured ...
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Interpolation: More advanced calculation than linear regression

I am currently modelling the customer growth of a company. I have assumed values for December 2020, 2021, 2022 and 2023 and want to estimate the values for the month in between. As you can see in the ...
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2 votes
1 answer
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Which is the covariance matrix between X ~ exponential (lambda) and X squared?

I need to find the covariance matrix between $X \sim exp(\lambda)$ and its square $X^2$. Can I interpret $X^n$ as an Erlang distribution (for which I can have the expectation/variance on Wikipedia :-) ...
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The random variable $log(\frac{X}{x_0})$ has an exponential distribution with parameter $\alpha$

It is said that a random variable $X$ has a Pareto distribution with parameters $x_0$ and $\alpha$ for $(x_0 > 0)$ and $(\alpha > 0)$ if $X$ has a continuous distribution for which the p.d.f. $f(...
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Bernoulli process and two exponentials

Suppose that a very long Bernoulli process gives a sequence with possible values: $A$ with probability $p$, and $B$ with probability $1-p$. The expected fraction of contiguous sequences of length $k$ ...
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CLT application to exponential distribution? [duplicate]

I'm a little confused how the CLT can apply to aggregations of the exponential distribution. It's my understanding that the CLT says, in plain English, "sample means from virtually any ...
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Calculating Average Days Between Customer Orders - Exponential Distribution?

I have detailed sales data including customer number, SKU, product line, and order date. For marketing purposes, I would like to know the average days between orders both on a per-customer, per ...
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1 vote
1 answer
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What is area under cumulative distribution represent? [duplicate]

Exponential distribution has following probability density function which explains the curvature of a line (For simplicity I am just going to work with x>=0): f(x) = lambda e^{-lambda*x} to find ...
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When can taking the following transformation of data $log_{10}(e^{mean(log(x))})$ be useful?

I wished to transform some data and have noticed that the geometric mean is a known transformation to normalise data with high differences in comparably large values relative to lower values. However, ...
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Consistency of $f( x| \theta ) = \exp(-(x- \theta ))$

Prove that the second smallest observation in a random sample of size n from following pdf is consistent estimator of $ \theta $ $$ f( x| \theta ) = \exp(-(x- \theta )) , \qquad x > \theta $$ ...
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1 answer
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Expected Value of Exponential CDF

I am given the following CDF and I want to calculate its expected value: $F(Y \leq y) =1-( 0.28e^{-0.5y} + 0.71e^{-0.25y})$ Creating the PDF: $f(Y \leq y) = \frac{71\mathrm{e}^{-\frac{x}{4}}+56\mathrm{...
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