Question: =+2.30. Let X1; X2; ... ; Xn be a random sample from the exponential distribution fX x2y 1 ex=2y; x50; y > 0, and let

=+2.30. Let X1; X2; ... ; Xn be a random sample from the exponential distribution fX ðxÞ¼ð2yÞ

1 ex=2y; x50; y > 0, and let the ordered X ’s be denoted by Y1 4Y2 4 4Yn. Assume that the underlying experiment is such that Y1 becomes available first, then Y2, and so on (for example, in a life-testing study) and that the experiment is terminated as soon as Y is observed for some specified r.

ðaÞ Show that the joint probability density function of Y1;Y2; ... ;Yr is

ð2yÞ

r n!

ðn rÞ!

exp

Pr i¼1 yi þ ðn rÞyr 2y

04y1 4 4yr < 1

ðbÞ Show that y1½

Pr i¼1 Yi þ ðn rÞYr has a chi-square distribution with 2r degrees of freedom.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Nonparametric Statistical Inference Questions!

Probability and Statistics - Problem Set c Keith M. Chugg October 2, 2015 1 Preliminaries, Combinatorics, Set Probability 1.1. A number of bats are in a cave. 2 bats can see out of their left eye. 3...

1. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her...

Exercises Chapter 2 2.1 Marginal and conditional probability: The social mobility data from Section 2.5 gives a joint probability distribution on (Y1 , Y2 )= (father's occupation, son's occupation)....

1. Let (X1, X2, ..., Xn) be a random sample from an exponential distribution with probability density function f(x) given by Ae-" for x > 0; f (ac) = 0 , otherwise. (a Find the moment estimator A of...

1. Let (X1, X2, ..., Xn) be a random sample from an exponential distribution with probability density function f(x) given by f (20 ) = for x > 0; 0 , otherwise. (a Find the moment estimator A of 1....

1. Let (X1, X2, ..., Xn) be a random sample from an exponential distribution with probability density function f(x) given by , for r > 0; f(I) = 0, otherwise. (a) Find the moment estimator A of 1....

Solve the following. 11. (a) Briefly explain the significance of the firm's cost of capital. What are the factors determining that cost of capital, and how can that cost be estimated? (b) Given that...

Let X1, X2, ..., Xn be a random sample from an unknown cumulative distribution function F supported on R. Describe and justify a method for estimating F(x), where x is an element in the support of F....

Please provide a detailed step by step explanation thank you 1. (a) Let {X1, X2, . . ., Xn} be a random sample of size n from the following probability density function: fx (I; a, 0) = (a - 1)!06 -le...

Let X1, X2,....., X100 be a random sample from a Pareto distribution with PDF f(x) = 3x^-4 if x ?1 0 otherwise. Find normal approximations for each of the following. 1. P(?100i=1 Xi > 160). 2. The...

Problem 2 (1 marks). Prove the following proposition. Proposition 1 Let U C R" be open. If f: U R is differentiable at x E U, there are positive real numbers and K such that ||u x||

When adding NaI and Acetone what products do you end up with for each of the substrates ? 1-Chlorobutane 2-Chlorobutane 2-Bromobutane t-butyl-Chloride (2-Chloro-2-methyl propane) Benzyl Chloride...

Which of the following must be proven by the plaintiff ( third party ) in cases of common law liability for fraud on the part of the auditor? Multiple select question. Improper use of the information...

Server farms such as Google and Yahoo! provide enough compute capacity for the highest request rate of the day. Imagine that most of the time these servers operate at only 60% capacity. Assume...