Question:

In this year's new student registration, UMY received 3000 students obtained from the test waves I, II and III. The following is the student data that is differentiated based on origin area: Based on the 3000 students, 70% came from Java, and the rest from outside Java. Of the 3000 students, 50% are from wave I. Meanwhile, of all students from Java, there are a total of 1300 students from batch I, and 400 students from wave II. Of the total number, students who came from batch III only 500 people.

Based on this data, do the following commands:

a. If one person is chosen randomly, what is the probability of being selected a student who comes from Java or from wave III?

b. If 2 consecutive people will be selected without return, what is the probability that a student will be selected from wave I and another from wave II?

Use the matrices P and D to construct a spectral decomposition of A = PDP - 1 9 0 O A= 0 : P = [ U , U2 Us] = NIm - Im NIM NIM NIM - IM - Im NIM NIM D=03 O 0 O A spectral decomposition of A is A = (Du, u, + (Du,u) + (Du,uj, where u, u! = . uzu) =, and ugu] =] (Simplify your answers.)4. Distributions associated with Poisson process (14 marks) Prove the following theorem: If W1, W2,..., are the occurrence times in a time-homogeneous Poisson process with. intensity > > 0, then conditioned on N(t) = n the random variables W1, W2. . .., W. have the joint probability density function f (w1. .... Wn) = nit-", for 0