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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Why the actual hazard ratio of the simulated time-to-event data is very different from the expected value?
I am trying to generate time-to-event data for two treatments, with 25 patients in each arm. I generated the survival time in the treatment arm using an exponential distribution with parameter 0.95, ...
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Hypoexponential distribution is stuck because of $\lambda_i \neq \lambda_j$?
The definition of the hypoexponential distribution (HD) requires that:
$$f(x)=\sum_i^d \left(\prod_{j=1,i\neq j}^{d}\frac{\lambda_j}{\lambda_j-\lambda_i}\right)\lambda_i e^{(-\lambda_ix)},\quad x>0
...
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Determining statistical significance on exponential variables and timeseries
I am attempting to determine statistical significance of various customer programs for farmers use of sustainable products. Farm sales of sustainable products follows roughly an exponential curve, ...
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Poisson Process: Probability distribution to describe time (distance) to successful event?
Given some length of time $t$ with successful events occurring in this interval at rate $\lambda$. Assume that only one successful event occurs during this interval of length $t$. Which distribution ...
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The natural parameterization of the exponential class of densities always exist?
Def. Exponential Class of Densities
The density function $f\left ( \mathbf{x};\mathbf{\Theta} \right )$ is a member of the exponential class of density functions iff
$$f\left ( \mathbf{x};\mathbf{\...
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Probability that one of two exponential random variables is the smaller
Let $X_1 \sim \text{exp} \left( {\lambda}_1 \right)$ & $X_2 \sim \text{exp} \left( {\lambda}_2 \right)$, and they are independent.
Now consider the random variable $Y = \min \left[X_1, X_2 \right]$...
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Convergence in probability of $n \cdot \min _{1 \leq j \leq n} X_j$?
Suppose $X_1, X_2, \ldots \sim U(0,1)$ and $X \sim \operatorname{Exp}(1)$, and $X_1,X_2, \dots , X$ are independent. Does it follow that $n \cdot \min _{1 \leq j \leq n} X_j \stackrel{P}{\rightarrow} ...
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Obtaining MLE of the parameter of exponential distribution
Let say I have a sample of size $n$ as $\{X_1,X_2,...,X_n\}$.
The sample points $X_i$ are integers, but each of them are actually integer ceiling of corresponding real number $\{Y_i\}$. For example if ...
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Find nth central moment for exponential distribution
I am trying to figure out what the nth central moment is for the exponential distribution.
Here is the formula for the nth moment:
$$
\mathop{\mathbb{E}}{[x^n]} =\dfrac{n!}{\lambda^n}
$$
My question: ...
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Calculate the distribution of the sum of mass arriving in random intervals over some period
Say we have a pipe spitting out potatoes.
The potato masses are distributed with exponential distribution
$$ P(\textrm{potato mass}= x)=\lambda_1e^{-\lambda_1x}$$
Each potato arrives with intervals ...
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Likelihood Function for the Two-Parameter Exponential Distribution with Interval-Censored Data
Suppose three similar items fail. For two of them we observe the exact time of failure: time 10 and time 12. For the third, all we know is that it failed between times 8 and 9, inclusive.
Suppose ...
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Density of sampled exponential data, with sampling weights proportional to x itself
Suppose $p(x) = \lambda e^{-\lambda x}$. However, our probability of observing a given sample of $x$ (denoted $z$) is further proportional to $x$ itself, i.e., $p(z\mid x) = \lambda e^{-\lambda x}$. ...
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Goodness-of-fit Tests
Continuing from my previous question here.
Furthermore, I intend to perform the chi-squared test and plot QQ-plots to test the hypothesis $H_0:\lambda=1$. I do not get to see the actual data though; I ...
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From Poisson to Erlang
If a customer arrives according to a Possion process with rate $\lambda$, how can I show that the time interval $X$ taken to receive $k$ customers is an Erlang-$k$ random variable with parameters $n$ ...
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Pre-compute random variable with same expected value and variance [closed]
Say I have an exponential distribution with parameter lambda λ.
Is there a way to generate 100-500 pre-computed values that when summed, have the same variance and mean as the exponential (and the ...
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Tandem system in queuing theory
We have a queue networking comprising of two queues $A$ and $B$ such that customers after finishing the service in $A$ go directly to $B$ then from $B$ go out of the network after service in $B$. ...
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Estimating the parameter $\beta$
The lifetime of computer monitors has a exponential distribution where the expected value can be written as:
$\mu(s) = \frac{\beta}{s}$
Where $s$ is how bright the monitor is and both $s$ and $\beta$ ...
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Link between the Beta and Exponential distribution
Let $n \geq 1$ be an integer. Let $X \sim \operatorname{Beta}(i, n - i + 1)$ where $i \in \{1, ..., n\}$. Therefore:
$$
X = \frac{A_n}{A_n + B_n}
$$
where
$$
A_n = \sum_{r = 1}^i Z_r, \qquad B_n = \...
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Fitting a GLM to data that resembles exponential decay
I'm trying to fit a generalized linear model to data that seems to have the behavior of exponential decay. The Y values, the response/endogenous variable describe expression, while there is just one X ...
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Calculating distribution of Poisson process at time t when a future value is known
Let $P$ be a Poisson point process with rate $\lambda$. If it is known that $P(t) = n$, how can we retroactively derive the conditional distribution of $P(k)$, where $k=t-s$ for $s<t$?
My idea: The ...
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Exponential and gamma distributions
I am trying to make a simulation of bacterium splits is exponentially distributed in R.
I am assuming that initially, bacterias have an exponential distribution. Let's say:
...
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Given independent random variables $X,Y$, and $M=\min(X,Y)$, what is $E(XM\mid Y=M)$?
Given independent random variables $X,Y$, and $M=\min(X,Y)$, what is $E(XM\mid Y=M)$ ?
The specific case I'm working on is assuming $X$ and $Y$ are exponential random variables with mean $\theta_X$ ...
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Find lambda for exponential distribution knowing a percentage
Suppose that rainfall duration has an exponential distribution. From regular measurements is known that 20% of rainfall ends within one hour. Calculate the mean value of rainfall duration.
I guess ...
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Parameter estimation on exponential distribution from a bounded subset of that distribution
I have a random variable that is exponentially distributed with some $\lambda$. I'm sampling observations from this variable, but I'm limited to observing only those that are less than some maximum ...
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How to apply exponentially decaying weights to converge PD estimates after 4-5 (or 7-10) years?
I have conditional Probability of Default (PD) estimates for 5 risk grades and for 6 year horizon using Markov Chain. I need to calibrate the first year to long run average and therefore change the ...
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Relationship among exponential distribution, sample size is n, allowed deviation from population mean is X%, then probability of this is %P
I performed monte carlo simulation for various sample sizes with exponential distribution.
I set "allowed deviation from population mean" as $5\%$. I iterated $1E5$ times for each sample ...
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Expected value of minimum of exponential random variables
I'm working on the following question:
A device contains two components, A and B. The lifespans of A and B are both exponentially distributed with expected lifespans of 5 years and 10 years ...
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Finding out variance of the sum of lengths of phone calls which have poisson distribution $(\lambda)$
Let N be the number of phone calls made by the customers of a phone company in a given hour. Suppose $N\sim Poisson(\beta)$, where $\beta > 0$ is known.
Let $X_i$ be the length of the i'th phone ...
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What an exponential distribution for a spatial poisson process answers
I use the Poisson distribution in virology where we try to answer: "What is the probability that X viruses enter a cell given a E(x)=MOI (=virus/cell)".
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How to perform calculations when the integral of a mixed exponential distribution pdf does not give 1?
Let Y be the time, for a mixture distribution with two exponential distributions, each multiplied by a and b and having 2 different parameters. How can calculations such as mean, variance and ...
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Why doesn't R use Inverse Transform Sampling to sample from the Exponential Distribution?
I was reading this question about the algorithm that R uses to sample from the Exponential($\lambda$) distribution. It looks like R uses the Ahrens-Dieter algorithm to sample from the exponential ...
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Question about how to interpret certain data (am I correct in using the exponential distribution)?
Let's say my data has certain parameters $x_{1},...,x_{n}$ and there are two events, let them be $h_{1},h_{2}$.
I'm considering interpreting the data via an exponential distribution because in this ...
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Why do simulated arrival times from a Poisson distribution seem to show periodicity?
I am experimenting with simulated arrival times drawn from a Poisson distribution. To construct the arrival times, I am randomly drawing inter-arrival times from the inverse CDF, which is ...
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modeling time between calls with exponential distribution
I've read that time between calls (in a call center) can be modeled with exponential distribution. My question is this: the shape of the exponential distribution has a decreasing nature. Suppose that ...
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Calculating the exponential growth rate against the standard deviation of the year coefficient
I have time-series abundance data for various locations. I would like to calculate the exponential growth rate for each location against the standard deviation of the year coefficient. My dataframe ...
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DataCamp exercise about distributions
I was studying some statistics in DataCamp and they assigned me this exercise that I can't solve. I tried speaking with people that know more statistics than me and we can't seem to agree in an answer....
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Difference between Power law distribution and Exponential distribution?
What is the difference between Power law distribution and Exponential distribution?
They both look similar!
kiran kumar
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Is log-rank parametric or non-parametric test, and why?
How is the log-rank test a "non-parametric test" according to wikipedia.org if one has to specify the parametric survival model for this test?
We may run log-rank under the assumption of ...
user318514
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Statistical Inference Casella & Berger Exercise 7.11 [duplicate]
I'm self studying statistics using Casella & Berger Statistical Inference and I'm confused about a detail in solution to exercise 7.11.
Here's the problem I'm try to solve:
Let $X_1, ..., X_n$ be ...
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Probability of $A<B$ when $A$, $B$ are random variable with different distribution?
Preparing exams, I ran into the following problem:
Edit: it shouldn't be represented as it was. Added the storkes.
Let $A$, $B$ be two independent variables having probability distribution:
$$
\...
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How to fit exponential and Poisson distributions to table of event times in R?
I have a vector of event dates where multiple events can occur on the same date. I have sorted these dates in chronological order and have also generated a table of times between events.
Here is an ...
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Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable
Consider a random variable
\begin{align}
U= a \log (Z+b)-Z
\end{align}
where $a,b>0$ and $Z$ is an exponential random variable.
Question: Can we find the variance of $U$?
Things that I tried
...
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Sum of iid Exponential observations then subtracting the minimum of the observations [duplicate]
Consider $n$ iid $X_1,...,X_n \sim Exp(1)$. My goal is to find the density of $\sum (X_i - X_{(1)})$.
My attempt
If we write out the entire summation in order statistics, we get
$X_{(1)}-X_{(1)} + X_{(...
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Exponential, Poisson or neither?
I have two variables in a marketing context: Advertiser spend per hour and conversions per hour. kernel density approximations of the underlying distributions look like below. Both distributions have ...
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Interpretation of drm parameter estimates and p-values for EXD.3 function in 'drc' package in R
I was wondering if someone could help me understand what the parameter estimates and p-values are saying in a three-parameter exponential decay function using the drm function in the 'drc' package in ...
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Goodness of fit for exponential distribution and large sample
I'm new to statistics and statistics are not my area of research so maybe my question is simple and answer is on the surface. In my research the empirical CDF for the data looks like exponential ...
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Exact/Approximate Confidence Interval of Parameter Ratio from two Samples of iid Exponential
Suppose 2 independent samples $X_1,...,X_n \sim Exp(\lambda_1)$ and $Y_1,...,Y_m \sim Exp(\lambda_2)$, and are iid within samples.
I am thinking about how to make an exact confidence interval for $\...
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MLE of parameters for a difference of two Exponential IID
Suppose I have $X_1 \sim Exp(\theta_1)$ and $X_2\sim Exp(\theta_2)$. Then it is not difficult to show that $Y = X_1 - X_2$ will have density:
$f_Y(y) = \frac{1}{\theta_1 + \theta_2}e^{-y/\theta_1}\...
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How to compute expectation of a exponentially distributed variable given the value of another variable?
I have 2 mutually independent random variables:
$s$ is distributed exponentially with parameter $\lambda$: $s\sim F(\cdot|\lambda)$
$\epsilon_x$ is distributed exponentially with parameter $\chi$: $\...
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How to computed "truncated shifted exponential distribution"?
I have a research problem to solve. For regular exponential distribution,
$$F(z|\lambda)=\begin{cases}0\;\;\;\;\text{if }z<0\\1-e^{-\lambda z}\;\;\;\;\text{ if }z\geq 0\end{cases}$$ with density $$...