Flashcards
Exponential Distribution and its Properties
Cards in this deck (26)
- 1Exponential Distribution with parameter λ>0. Its probability density function is given by
- 2Exponential Distribution with parameter λ>0. Its probability density function is given byhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_161834-16A2872031A407630B6.png
- 3Exponential Distribution with parameter λ>0. Its probability cumulitive function is given byhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_161925-16A2872B89B660B61DF.png
- 4The mean of the exponential distribution, E[X], is given byhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_162041-16A28740A581F8A615F.png
- 5The moment generating function φ(t) of the exponential distribution is given byhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_162154-16A287500E93DF3F0EB.png
- 6All the moments of X can now be obtained by differentiating, so E[X2]https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_162332-16A2876B449358A15DE.png
- 7Variance of Exponential Distribution ishttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_162435-16A2877A9214ACE57D5.png
- 8A random variable X is said to be without memory, or memoryless, ifhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_134557-16A287861BC043EF13A.png
- 9A random variable X is said to be without memory, or memorylesshttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_134706-16A2879B7BE68AC486A.png
- 10Example 5.2 Suppose that the amount of time one spends in a bank is exponentially distributed with mean ten minutes, that is, λ = 1/10.What is the probability that a customer will spend more than fifteen minutes in the bank?If X represents the amount of time that the customer spends in the bank, then the first probability is just
- 11Example 5.2 Suppose that the amount of time one spends in a bank is exponentially distributed with mean ten minutes, that is, λ = 1/10. What is the probability that a customer will spend more than fifteen minutes in the bank given that she is still in the bank after ten minutes?question asks for the probability that a customer who has spent ten minutes in the bank will have to spend at least five more minutes. However, since the exponential distribution does not “remember” that the customer has already spent ten minutes in the bank, this must equal the probability that an entering customer spends at least five minutes in the bank. That is, the desired probability is just
- 12Example 5.3 Consider a post office that is run by two clerks. Suppose that when Mr. Smith enters the system he discovers that Mr. Jones is being served by one of the clerks and Mr. Brown by the other. Suppose also that Mr. Smith is told that his service will begin as soon as either Jones or Brown leaves. If the amount of time that a clerk spends with a customer is exponentially distributed with mean 1/λ, what is the probability that, of the three customers, Mr. Smith is the last to leave the post office? TermConsider the time at which Mr. Smith first finds a free clerk. At this point either Mr. Jones or Mr. Brown would have just left and the other one would still be in service. However, by the lack of memory of the exponential, it follows that the amount of time that this other man (either Jones or Brown) would still have to spend in the post office is exponentially distributed with mean 1/λ. That is, it is the same as if he were just starting his service at this point. Hence, by symmetry, the probability that he finishes before Smith must equal ½.
- 13Let X1, . . . , Xn be independent and identically distributed exponential random variables having mean 1/λ. It follows from the results that X1 +…+Xn has a gamma distribution with parameters n and λ. Because there is nothing to prove when n= 1, let us start by assuming that X1+…+Xn−1 has density given byhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_140825-16A28800DE4217D6542.png
- 14X1+…+Xn−1+Xn has density given byhttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_140832-16A2881113A249DBC62.png
- 15Suppose that X1 and X2 are independent exponential random variables with respective means 1/λ1 and 1/λ2; what is P{X1https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_141005-16A2881BC4D4FE5F12E.png
- 16Suppose that X1,X2, . . . , Xn are independent exponential random variables, with Xi having rate μi, i = 1, . . . , n. It turns out that the smallest of the Xi is exponential with a rate equal to the sum of the μi .https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_141135-16A28824E6E3D077AFF.png
- 17Let X1, . . . , Xn be independent exponential random variables, with respective rates λ1, . . . , λn. A useful result, generalizing P[X1where the final equality uses along with the fact that min(i≠j)=Xj is exponential with rate ∑(i≠j)λj.
- 18Another important fact is that min_i Xi and the rank ordering of the Xi are independent.Because mini Xi >t means that all the Xi are greater than t, it follows from the lack of
memory property of exponential random variables that their remaining lives beyond t remain independent exponential random variables with their original rates.
Definition - 19Example 5.8 Part1 Suppose you arrive at a post office having two clerks at a moment when both are busy but there is no one else waiting in line. You will enter service when either clerk becomes free. If service times for clerk i are exponential with rate λi, i = 1, 2, find E[T], where T is the amount of time that you spend in the post office.TermLet Ri denote the remaining service time of the customer with clerk i, i = 1, 2, and note, by the lack of memory property of exponentials, that R1 and R2 are independent exponential random variables with respective rates λ1 and λ2. Conditioning on which of R1 or R2 is the smallest yields
- 20Now, with S denoting your service timeThe final equation used that conditional on R1
- 21https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_144924-16A288977F926AC7F4D.pngAs we can show in a similar fashion that
- 22We obtain the resultWe obtain final result
- 23Another way to obtain E[T] is to write T as a sum, take expectations, and then condition where needed. This approach yieldshttps://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_145232-16A288C599431D92C05.png
- 24https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_145232-16A288CCD714EF83932.pngTo compute E[S], we condition on which of R1 and R2 is smallest.
- 25Convolutions of two Exponential Random Variables.https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_151536-16A288E6C4616D35421.png
- 26Convolutions of Exponential Random Variables.https://s3.amazonaws.com/classconnection/529/flashcards/17745529/png/screenshot_2019-04-16_151551-16A288F41A77A169182.png
Information
- Description
- Flashcards on Exponential Distribution and its Properties, created by Anelia Mynzhassarova on 16/04/2019.
- Created by
- Anelia Mynzhassarova
- Views
- 2
- Rating