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. 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 Standard error of the coefficient of variation

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

38. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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...