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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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
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...
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...
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)
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?
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...
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...
$$f(y) = \begin{cases} \int_0^y\frac1\beta e^{\frac {-t}\beta}dt = -e^{\frac {-y}\beta}+1 & \text{for } 0 ≤ y < ∞,\\ 0& \text{for } elsewhere\end{cases}$$
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...
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...
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...
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...
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...
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)
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...
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...
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...
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...
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...
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...
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...