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. J

    Exponential distribution problem

    Hello forum, I recently lost my notes on this matter, so I hope someone can fill in the gaps in my memory. The problem I am working on is the following: For one ticket window, the waiting time for one people satisfies an exponential distribution λe-λt and expected waiting time is 4 min...
  2. C

    Integral of exponential distribution from zero to infinity

    Homework Statement here's a problem from my assignment let integral p(x)dx=Ae^-(x/a) dx...(1) find value of A, that makes integral p(x)dx=1; and find mean x so that integral x*p(x)dx=a...2) Homework Equations now to solve the first one i found out A to be (-1/a*e^(x/a)) but...
  3. M

    Gamma and exponential distribution

    Homework Statement prove that, for gamma distribution μ=αθ σ^2=αθ^2 for exponential distribution μ=θ σ^2=θ^2
  4. E

    A little help understanding the exponential distribution

    What exactly is the difference between the mean waiting time and the median waiting time for an exponential distribution? I'm looking for a slightly intuitive understanding. I know the formulae, with the mean waiting time as 1/λ and the median as ln2/λ (which I notice is also the formula for...
  5. S

    MLE of P(X<2) - Exponential distribution

    Homework Statement Find the MLE of θ = P (X≤ 2) in a random sample of size n selected from an exponential distribution EXP(λ) Homework Equations f(x, λ) = λ e^(-λx) F(x, λ) = 1 - e^(-λx) The Attempt at a Solution I know how to find the MLE of the mean of an exponential...
  6. Mentallic

    Maths - Solving for the CDF of Y=\sqrt{X}

    Homework Statement If we have the exponential distribution f_X(x)=\frac{1}{2}e^{-x/2} then show that the cumulative distribution function of Y=\sqrt{X} is given by F_Y(y)=1-e^{-y^2/2}Homework Equations F_Y(y)=f_X(x)\cdot\left| \frac{dx}{dy}\right| F_Y(y)=f_X(h^{-1}(y))\cdot\left|...
  7. M

    Stats/Probability: Joint Exponential Distribution

    Homework Statement Suppose that X=time to failure for a component has an exponential distribution with lambda =.25. Suppose that 9 of the components are selected and their failure times noted. Compute the probability that 3 of the components fail between times 1 and 2, and 4 of the components...
  8. B

    Exponential Distribution and radio active decay

    Typical radio active decay question Half time = 1 year λ = ln 2 here Q 1: if we have 1024 atoms at t=0, what is the time at which the expected number remaining is one. Easy, I get 10 years Q 2: The chance that in fact none of the 1024 atoms remains after the time calculated in c...
  9. S

    Exploring the Exponential Distribution: Generating Function Method

    Homework Statement F_{X}(x)= λe^{-λx} \;for\; x>0 \;\;\;and \;0 \;otherwise After finding the characteristic function for the Exponential Distribution, which is (I could do this without problem); F_{X}(k)=λ(λ-ik)^{-1} Now the question is; Let X_1,X_2,\ldots,X_i be i.i.d. exponential...
  10. D

    Probability question about exponential distribution

    Homework Statement A manufacturing process produces 92% good chips (G) and 8% bad chips (B). The lifetime, in seconds, of chips is exponentially distributed E(\lambda). For good chips, \lambda1=1/20000 For bad chips,\lambda2=1/1000 Every chip is tested for 50 seconds prior to leaving the...
  11. B

    Distribution of the sample mean of an exponential distribution

    Let's say you have a random sample of 5 values that are drawn from an exponential distribution with a mean of 8. How do I find the distribution of Ybar, which is the sample mean of the 5 random variables? [Note: Ybar = 1/5(Y₁+Y₂+Y₃+Y₄+Y₅)] I know that for an exponential distribution with...
  12. S

    Help with exponential distribution

    X1,X2,...,XN are independently identically exponentially distributed with expected value of 5. How can I compute X[bar]n when n=20 and N=1000? Then compute the proportion of values of X[bar]n that lie between 6.99 and 7.01. repeat the above question with n=100 My thoughts so basically...
  13. R

    Me the variance of the score function in exponential distribution

    please help me! the variance of the score function in exponential distribution Homework Statement My question is about exponential function, with its density function known as f(x;theta) = (1/theta) e^(-x/theta) for all x>0. where E(x) = theta, var(x) = theta^2 My question is, what...
  14. E

    Exponential Distribution and median

    Homework Statement IF X has an exponential distribution with parameter \lambda, derive a general expression for the (100p)th percentile of the distribution. Then specialize to obtain the median. Homework Equations X.exp(\lambda)=\lambda e^{-\lambda x} for x>0 p=F(\eta (p)) =...
  15. M

    Conditional expectation of exponential distribution.

    I have been stuck at this calculation. There are two exponential distributions X and Y with mean 6 and 3 respectively. We need to find E[y-x|y>x] I keep getting it negative, which is clearly wrong. Anybody wants to try it?
  16. A

    Bus simulation in excel, exponential distribution departure

    Homework Statement The bus service operates between points A and B. The buses depart from the terminal station A with headways that follow the negative exponentioal distributiona with parameter 10 trips per hour. Assume that the characteristics of the system remain the same over time (not...
  17. S

    Exponential Distribution Problem

    I am having trouble solving this problem. I'm not sure how to solve this problem... Assume X and Y are independent exponential random variables with means 1/x and 1/y, respectively. If Z=min(X,Y): Is Z exponentially distributed as well (if so, how do you know)? What is the expectation of Z...
  18. A

    Reliability question with exponential distribution

    Q. A certain type of memory chip is known to have an exponential life distribution with a failure rate of 0.15*10^-5. a) What is the probability that a memory chip will survine 20,000 hours of use? b) What is the probability it will fail in the next 35,000 hours if it has survived 20,000...
  19. 6

    Exponential Distribution and probability

    Homework Statement The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes. What is the probability that there are more than three calls in one-half hour? Homework Equations F(x) = P(X <= x) = 1 - e^-(lamba*x) The Attempt at a Solution...
  20. M

    Exponential distribution likelihood

    In general what do you do when some kind of data is missing from some data that follows exponential distribution. For example, say 3 observations are made by an instrument where x1=5, x2=3, but for x3 the instrument can not give a specific answer because it can't measure past 10. So the only...
  21. M

    Likelihood Function - Exponential Distribution

    Homework Statement X is exponentially distributed. 3 observations are made by an instrument that reports x1=5, x2=3, but x3 is too large for the instrument to measure and it reports only that x3 > 20 . (The largest value the instrument can measure is 10) a)What is the likelihood function...
  22. A

    What is the Probability of Two Major Cracks in a 10 Mile Stretch of Highway?

    Homework Statement The distance between major cracks in the highway follows an exponential distribution with a mean of 5 miles. What is the probability that there are two major cracks in a 10 mile stretch of the highway? Homework Equations exponential dist: f(x) = Le(-Lx) where L...
  23. S

    I don't understand the exponential distribution at all

    This is driving me completely crazy! QUESTION 1: There are two interpretations I find for the exponential distribution: 1) It models the lifetime of something that does not age in the sense that the probability of functioning yet another time unit does not depend on its current age. So...
  24. E

    Joint Bivariate Exponential Distribution

    Homework Statement Consider two components and three types of shocks. A type 1 shock causes component 1 to fail, a type 2 shock causes component 2 to fail, and a type 3 shock causes both components 1 and 2 to fail. The times until shocks 1, 2, and 3 occur are independent exponential random...
  25. R

    What is the Cost of Lifetime for an Exponentially Distributed Component?

    Homework Statement component has lifetime X that is exponentialy distibuted with parameter gamma. a) if the cost per unit time is a constant, c, what is the expectec cost of its lifetime? b) if c is not constant, and=c(1-.5e^(a*x) such that a<0. (aka it costs more over time) what is its...
  26. T

    Multivariate Exponential distribution

    Hi to all :) Does anyone have any idea how the expression for a multivariate exponential distribution looks like? If possible, can you post the source url? Commonly available is the multivariate normal distribution. thanks in advance :biggrin:
  27. J

    Exponential growth and exponential distribution

    I'm a bit confused about exp growth and exp distribution. Suppose I have a branching process situation, where there are n individuals in generation 0, and each individual produces a random number of offspring, according to some distribution (say Poisson), at each generation. Now, then after a...
  28. B

    Probability - Exponential Distribution

    Here's the evil question: Let X~Exponential(alpha). Derive and name the pdf of Y=(alpha)X
  29. Z

    Characteristic function of an exponential distribution

    I need to calculate the characteristic function of an exponential distribution: \phi _X \left( t \right) = \int\limits_{ - \infty }^\infty {e^{itX} \lambda e^{ - \lambda x} dx} = \int\limits_{ - \infty }^\infty {\lambda e^{\left( {it - \lambda } \right)x} dx} I have arrived at the...
  30. C

    Calculating Probability for Exponential Distribution in Unplanned Shutdowns

    I am having a lot of trouble with a homework question from my book. It asks: The time between unplanned shutdowns of a power plant has an exponential distribution with a mean of 20 days. Find the probability that the time between two unplanned shutdowns is more than 21 days. I know this...
  31. D

    Attempts until Rnd<constant, has exponential distribution?

    Suppose a number U is generated from an uniform distribution [0,1]. If you repeat the process until U < some constant, does the number of loops have an exponential distribution? If so, could you point the way to a proof? Thanks in advance.
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