Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [exponential-distribution]

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

For an Exponential distribution as a probability density function:

$f(x;\lambda) =\lambda e^{-\lambda x}\quad$ for $x \ge 0 $

and

$f(x;\lambda) =0\quad$ for $x \lt 0 $

where $\lambda$ is the rate parameter.

1538 questions
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4 answers

Pdf of the difference of two exponentially distributed random variables

Suppose we have two independent random variables $Y$ and $X$, both being exponentially distributed with respective parameters $\mu$ and $\lambda$. How can we calculate the pdf of $Y-X$?
Mathematics Lover
  • 725
30
votes
2 answers

How to prove that minimum of two exponential random variables is another exponential random variable?

How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y)
user82004
28
votes
5 answers

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 relevant distribution. What is the technique or…
Ramamurthy Shankar
  • 289
25
votes
1 answer

Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function. Well, I know that $P(T_i>t)=e^{-\lambda…
TI Jones
  • 541
16
votes
2 answers

Characteristic function of exponential and geometric distributions

I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you please guide me on getting them? Here is my solution so far: Characteristic function of exponential distribution: $\phi(t) =…
user48405
15
votes
3 answers

How can I prove both series are equal?

Given $n$ random variables $X_1, X_2,\dots, X_n$, which are independent and exponentially distributed with rate parameter $\lambda$, I was able to prove that $$\mathbb E[\max\{X_1, X_2,\dots, X_n\}] = \frac{1}{\lambda}\sum_{k=1}^n(-1)^{k-1}\frac{n…
Victor Durojaiye
  • 169
15
votes
2 answers

On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will type it out here as well. I would like to get some…
jlcv
  • 1,132
13
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3 answers

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 manually) I get $9$, so I am slightly confused. I…
Szymon Brych
  • 321
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2 answers

What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0 \\ 0 & x <…
BLAZE
  • 1
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votes
2 answers

Is the average corner a curve?

Consider the random experiment in which $y_{-1}, y_{+1}$ are i.i.d. exponential random variables with rate parameter $\lambda$, sampled at $x=-1$ and $x=1$, respectively. For each sample, consider the lines passing through the points $(-1, y_{-1})$…
sam wolfe
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9
votes
1 answer

Finding UMVUE of $\theta$ when the underlying distribution is exponential distribution

Hi I'm solving some exercise problems in my text : "A Course in Mathematical Statistics". I'm in the chapter "Point estimation" now, and I want to find a UMVUE of $\theta$ where $X_1 ,...,X_n$ are i.i.d random variables with the p.d.f $f(x;…
user88914
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8
votes
1 answer

Prove that if $X\sim\mathcal{N}(0,1)$, then $X^2-1$ is subexponential

Definition: A random variable $X$ with $E[X]=0$ is called sub-exponential with parameters $(\nu,\alpha)$ iff for each $\lambda$ satisfying $|\lambda|<\frac{1}{\alpha}$, we have: $$E[e^{\lambda X}]\le e^{\frac{\lambda^2 \nu^2}{2}} $$ We write this as…
Arman Malekzadeh
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8
votes
1 answer

The time it takes for a candle having a lifetime that follows an exponential distribution to go off

We have $5$ candles each having a lifetime which follows an exponential distribution with parameter $\lambda$. We light up each candle at time $t=0$. Assume that $Y$ is the time that it takes for the third candle to go off. What is the expectation…
Jared Garreta
  • 81
8
votes
2 answers

The expectation of $e^X \left(1-(1-e^{-X}\right)^n)$ when $X$ has Exponential Distribution

To my surprise, I was able to evaluate the following expression in Mathematica: $$E\left[e^X \left(1-(1-e^{-X}\right)^n) \right] = \frac{y}{y-1} \left(1-\frac{1}{\binom{n+y-1}{y-1}}\right)\quad X\sim\text{Exp}(y)$$ with the right hand side being…
Thomas Ahle
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8
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0 answers

First moments of Geometric Brownian Motion-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^-(0,1,\lambda)$$ where $\operatorname{EMG}^-$ is…
sfortney
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