What is Exponential distribution: Definition and 81 Discussions

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.

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  1. S

    Finding probability related to Poisson and Exponential Distribution

    My attempt: (i) ##\lambda =3## (ii) (a) ##P(N_{2} \geq 1=1-P(N_{2} =0)=1-e^{-6} \frac{(-6)^0}{0!}=0.997## (b) ##P(N_{4} \geq 3)=1-P(N_{4} \leq 2)=0.999## (c) ##P(N_{1} \geq 2) = 1-P(N_{4} \leq 1)=0.8## Do I even understand the question correctly for part (i) and (ii)?(iii) The expectation of...
  2. A

    A Standard error of the coefficient of variation

    What is the standard error of the coefficient of variation in an exponential distribution?
  3. U

    MHB How to solve an expression with inverse of exponential distribution

    I have an Energy harvesting expression something like the following $R = \tau B \log\Big(1 + \frac{E h^2}{\tau r^\alpha\sigma^2} \Big)$ $E = \tau(2^{R/\tau B}-1 )\frac{r^\alpha\sigma^2}{h^2}$ Let all constant terms as $a$ to simplify the expression into : $E = a\frac{1}{h^2}$ $E$ is a random...
  4. M

    Mean and var of an exponential distribution using Fourier transforms

    Hi, I was just thinking about different ways to use the Fourier transform in other areas of mathematics. I am not sure whether this is the correct forum, but it is related to probability so I thought I ought to put it here. Question: Is the following method an appropriate way to calculate the...
  5. archaic

    Exponential distribution probability exercise

    1) Since I want at least ##6## flights to come within ##2## hours, then the time interval between each should be, at worse, ##2/6=1/3## hours, and the probability is ##P(X\leq1/3)=1-e^{-1/3}##. 2) The probability that at best 5 airplanes arrive at the airport is...
  6. WMDhamnekar

    MHB Exponential distribution question

    Hi, I want to know how the highlighted steps are arrived at in the first page. What are \(R_X (y), R'_X (y),F'_X (0) ? \)How \(R_X (0) = 1 ?\) Solution to differential equation should be \(R_X (y)=K*e^{\int{R'_X (0) dx}}\) But it is different. How is that? What is $-R'_X...
  7. L

    I Birth and death process and 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 individual the time until the next infection follows an exponential distribution with expected value 1/λ...
  8. R

    I Re-scaling of exponentially distributed numbers

    For simplicity, let ##N=1##. The following histograms show my results. The generated random numbers are initially exponentially distributed. But after re-scaling they become almost uniformly distributed. What is the cause of that, and is there a solution? P.S. Here is my code in Matlab...
  9. user366312

    I Probability equation in case of an exponential distribution

    Why is ##P(X>5|X>1) = P(X>4)## in case of an exponential distribution? Can anyone kindly explain it to me?
  10. Livio Arshavin Leiva

    A What exactly is a "rare event"? (Poisson point process)

    These days I've been reading in the internet about the Poisson Distribution because that was a concept I couldn't manage to understand completely when I studied it, so since then I've been always quite curious about Poisson processes, and how there are a lot of natural phenomena (mostly the...
  11. M

    MHB Exponential distribution - inequality

    Hey! :o We consider the exponential distribution. I want to show that $$\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right )\geq \frac{\lambda^4-1}{\lambda^4}$$ I have shown so far that \begin{align*}\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right...
  12. M

    MHB Normal and exponential distribution

    Hey! :o I am looking at the following: 1) A machine produces $100$ gram chocolate. Due to random influences, not all bars are equally heavy. From a long series of observations it is known that the mass X of a chocolate is distributed normally with parameters $\mu = 100$g and $\sigma = 2.0$g...
  13. O

    Exponential Distribution, Mean, and Lamda confusion

    Homework Statement Accidents at a busy intersection follow a Poisson distribution with three accidents expected in a week. What is the probability that at least 10 days pass between accidents? Homework Equations F(X) = 1- e-λx μ = 1/λ The Attempt at a Solution Let x = amount of time between...
  14. C

    I Exponential Distribution Question

    Hi all, Can anyone teach me this problem ? Thanks The life of a tiger is exponentially distributed with a mean of 15 years.If a tiger is 10 years old, what is the expected remaining life of the tiger? A 5 years B 10 years C 15 years D Longer than 15 years
  15. iikii

    Long-run proportion in state ′A′

    The question asks: A physical device can be in three states: A,B,C. The device operates as follows (all time units are in hours): The device spends an exponentially distributed amount of time in stateAA (with mean of 12minutes) and then with probability 0.6 goes to state B, and with...
  16. iikii

    Computer Server Down Probability

    So the problem asks: A computer server runs smoothly for Exp(0.2) days and then takes Exp(0.5)days to fix. The server is running fine on Monday morning, t=0. Find the probability that the server was fixed at least once (i.e. at least one complete repair was done) in the next 7 days and the...
  17. L

    Radioactive decay, relation between binomial to expon. dist

    You can model the probability for radioactive decay as a Poisson distribution. This is the probability for radioactive decay within a specific time interval. (I probably got some of it wrong here). P(k,μ)=λ^k⋅exp(-μ)/k! Is there a way to use this formula to derive the other formula for...
  18. T

    Expected value and variance of multivariate exponential distr.

    Homework Statement https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png I want to find variance matrix and expected variance vector of Y=(Y1,Y2). Y1 and Y2 are independant. Γ is the gamma function and ϒ is a known parameter. λ1>0 λ2>0 and ϒ>0...
  19. T

    Distribution of exponential family

    Let's say my probability function is given by: p(y1,y2)=Γ(y1+y2+γ)/((y1+y2)!*Γ(γ)), when γ>0 is known. I suppose it is from an exponential family but I can't write in canonical form because I'm only familiar with exponential family with one variable so I'm confused now when there's to variable...
  20. T

    Exponential distribution problem

    Homework Statement The amount of time that a surveillance camera will run without having to be reset is a random variable having the exponential distribution with beta = 50 days. find the probabilities that such a camera will a) have to be reset in less than 20 days b) not have to be reset in...
  21. S

    MHB What are the Calculations for Exponential Distribution in Bank Arrival Times?

    Let $X$ = the time between two successive arrivals at the drive-up window of a local bank. $X$ has an exponential distribution with $\lambda = 2$. That is the probability density of $X$ is $f(X | \lambda) = \lambda e^{-\lambda x}, X > 0 $ with $\lambda = 2$. Compute the following: a) The...
  22. A

    MATLAB MatLab: array of numbers unequal distribution

    I want to create an array of numbers between 0 and 0.1 where the points are clustered around an arbitrary point x1 (0 < x1 < 0.1). I want the points to get exponentially closer together near x1 from either side and and get further apart towards the outer limits. I am using MatLab and was trying...
  23. L

    Poisson process and exponential distribution arrival times

    Homework Statement Customers arrive in single server queue to be serviced according to Poisson process with intensity 5 customers an hour. (a) If the customers begin to arrive at 8am, find the probability that at least 4 customers arrived between 9am and 10am. (b) Find the probability that the...
  24. H

    Exponential Distribution Probability

    Homework Statement The life times, Y in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.6 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if...
  25. S

    Determine probabilities involving exponential distribution

    Homework Statement Problem(s): Suppose that X has an exponential distribution with mean equal to 10. Determine the following: (a) P(X > 10) (b) P(X > 20) (c) P(X < 30) (d) Find the value of x such that P(X < x) = 0.95. Correct answers: (a) 0.3679 (b) 0.1353 (c) 0.9502 (d) 29.96 Homework...
  26. Greg Bernhardt

    What is exponential distribution

    Definition/Summary The exponential distribution is a probability distribution that describes a machine that it equally likely to fail at any given time. Equations f(t) = e^{-\lambda t} \lambda Extended explanation A machine is equally likely to fail at any given time. For any t...
  27. D

    MHB How Do You Calculate the Probability in an Exponential Distribution?

    Help? Suppose the random variable Y has an EXP(2) distribution. What is P(Y > 1)? (Round to four decimal places as appropriate.)
  28. O

    Exponential distribution question

    Hi. I notice that some values of X on the exponential distribution PDF have a value of around 1. I understand the integral ends up being one, since those values of X are less than 1. But P(X) at those points still gets to 1, or thereabouts. How does that make sense, that the probability of a...
  29. O

    Standard deviation in exponential distribution

    What is the significance of the standard deviation (equal to the mean) in an exponential distribution? For example, as compared to the standard deviation in the normal distribution, which conforms to the '68-95-99.7' rule? thanks
  30. K

    Testing/proving X-bar oof an exponential distribution

    ok, so I have a list of students with GPA, I checked the probability plot and I think its a Exponential distribution, take a look: So I am given a χ-bar to prove, and I have to prove or test it with three different types of test, I don't know which ones or how to do them in miniTab...
  31. T

    Gamma distribution from sample mean of Exponential distribution

    Homework Statement Let X1, X2,...,Xn be a random sample from the exponential distribution with mean θ and \overline{X} = \sum^{n}_{i = 1}X_i Show that \overline{X} ~ Gamma(n, \frac{n}{θ}) Homework Equations θ = \frac{1}{λ} MGF Exponential Distribution = \frac{λ}{λ - t} MGF Gamma...
  32. Y

    MHB Calculating the Density Function for X/Y with Exponential Distributions

    X,Y r.v statistically independent ,with exponential Distribution. calculate the density function of X/Y (Let $X$ have distribution ${\lambda}e^{-{\lambda}x}$ and $Y$ have distribution ${\lambda}e^{-{\lambda}y}$ i know i should use transformtion u=X+Y ;v=X/Y to solve it)
  33. T

    Question on exponential distribution?

    Homework Statement Homework Equations f(x) = e-λλx/x! The Attempt at a Solution Initially I thought I could solve this problem using the Law of Memoryless. That, the solution would just be P(X <= 2). However, I was wrong. Turns out the solution is P(X <= 4.5) - P(X<= 2.5). Does anyone know why?
  34. O

    Exponential distribution, memory

    I am told that an exponential distribution is memoryless. But why aren't other distributions, such as the normal distribution, also memoryless? If I pick a random number from an exponential distribution, it is not effected by previously chosen random numbers. But isn't that also the case for...
  35. T

    Derivation of the exponential distribution - that infinitesimal

    Hello, I've been looking at the derivation of the exponential function, here http://www.statlect.com/ucdexp1.htm amongst other places, but I don't get how, why or what the o(delta t) really does. How does it help? It's really confusing me, and all the literature I've looked at just...
  36. A

    Exponential Distribution with Probability

    $$f(y) = \begin{cases} 0& \text{for }y< 0,\\ 2y& \text{for }0 ≤ y ≤ .5,\\ 6-6y& \text{for }0.5 < y ≤ 1, \\0& \text{for } y > 1\end{cases}$$ (1) Find cumulative distribution function, F(y) $$F(y) = \begin{cases} 0& \text{for }y< 0, \\\int_0^y 2t dt = y^2 & \text{for } 0 ≤ y ≤ .5,\\.5^2+...
  37. A

    Exponential Distribution with Probability

    $$f(y) = \begin{cases} \int_0^y\frac1\beta e^{\frac {-t}\beta}dt = -e^{\frac {-y}\beta}+1 & \text{for } 0 ≤ y < ∞,\\ 0& \text{for } elsewhere\end{cases}$$ P(Y>3) = 1 - P(Y ≤ 3) = 1 - (-e^{-3/beta}+1) = .1353 When I take log to both sides, I get 3.453. When I take ln to both sides, I get...
  38. A

    Exponential distribution moment generating function to find the mean

    With mean = 2 with exponential distribution Calculate E(200 + 5Y^2 + 4Y^3) = 432 E(200) = 200 E(5Y^2) = 5E(Y^2) = 5(8) = 40 E(4Y^3) = 4E(Y^3) = 4(48) = 192 E(Y^2) = V(Y) + [E(Y)]^2 = 2^2+2^2= 8 E(Y^3) = m_Y^3(0) = 48(1-2(0))^{-4} = 48 is this right?
  39. A

    Application for exponential distribution

    The amount of time to finish a operation has an exponential distribution with mean 2 hours Find the probability that the time to finish the operation is greater than 2 hours. My thinking is to integrate the exponential probability function. After integrating it, I got -e^{-y/2} + 1 , 0 ≤ y...
  40. G

    Exponential Distribution and Waiting Time

    Homework Statement Suppose that the waiting time for the CTA Campus bus at the Reynolds Club stop is a continuous random variable Z (in hours) with an exponential distribution, with density f(z) = 6e–6z for z ≥ 0; f(z) = 0 for z < 0. (a) What is the expected waiting time in minutes (the...
  41. G

    Estimate exponential distribution parameter

    Homework Statement A certain type of transistor has an exponentially distributed time of operation. After testing 400 transistors, it is observed that after one time unit, only 109 transistors are working. Estimate the expected time of operation. Homework Equations The Attempt...
  42. S

    Exponential distribution, two exercises

    Homework Statement Waiting time in a restaurant is exponentially distributed variable, with average of 4 minutes. What is the probability, that a student will in at least 4 out of 6 days get his meal in less than 3 minutes? Homework Equations The Attempt at a Solution If I...
  43. P

    Is λ Equal to 0.01 or 100 for a Light Bulb with 100 Hour Life Expectancy?

    Hi, Homework Statement If the life expectancy of a light bulb is a random exponential variable and equal (on average) to 100 hrs, is λ then equal to 0.01 or to 100? (λ = 1/expectation)
  44. Mentallic

    Exponential Distribution memory loss

    Homework Statement Show that the exponential distribution has the memory loss property. Homework Equations f_T(t) = \frac{1}{\beta}e^{-t/\beta} The memory loss property exists if we can show that P(X>s_1+s_2|X>s_1) = P(X>s_2) Where...
  45. M

    Conditional exponential distribution and exponential evidence

    Homework Statement This is a subset of a larger problem I'm working on, but once I get over this hang up I should be good to go. I have a set of measurements x_n that are exponentially distributed p(x_n|t)=e^{-(x_n-t)} I_{[x_n \ge t]} and I know that t is exponentially distributed as...
  46. G

    Probability Question - Exponential Distribution

    Homework Statement Suppose that X has an exponential distribution with mean μ. Find the probability that x lies within one standard deviation of its mean, that is find P(μ-σ≤X≤μ+σ) Homework Equations The Attempt at a Solution If I'm not mistaken the standard deviation is equal...
  47. H

    Probability question involving exponential distribution.

    Homework Statement The total claim for a health insurance policy follows a distribution with density function f(x) = (1/1000)e^{-x/1000}, x>0 The premium for the policy is set at 100 over the expected total claim amount. If 100 policies are sold, what is the approximate probability that...
  48. S

    Exponential distribution word problem

    The Information Systems Audit and Control Association surveyed office workers to learn about the anticipated usage of office computers for holiday shopping. Assume that the number of hours a worker spends doing holiday shopping on an office computer follows an exponential distribution. a) The...
  49. S

    PDF of an exponential distribution

    Homework Statement Hi! I'm trying to find the PDF of W = abs(X-λ), where X is an exponential R.V. with rate parameter λ>0. Homework Equations The PDF for an exponential distribution is ∫λe^(-λx)dx. Taking the derivative of a CDF will yield the PDF for that function (I'm aware there are...
  50. D

    Proving the memoryless property of the exponential distribution

    Given that a random variable X follows an Exponential Distribution with paramater β, how would you prove the memoryless property? That is, that P(X ≤ a + b|X > a) = P(X ≤ b) The only step I can really think of doing is rewriting the left side as [P((X ≤ a + b) ^ (X > a))]/P(X > a). Where...