# Questions tagged [exponential-distribution]

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

671 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $\{N_2(t), t \ge 0\}$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$f_X(x)=\frac{2x}{\alpha}e^{-x^2/{\alpha}}.$$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
### 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 ...