The condition of an interaction changes every day as per a two-state Markov chain. In the event that the cycle is in state I during one day, it is in state j the next day with likelihood Pi, 1, where

Pao = 0.4, P0.1 = 0.6, P,0 = 0.2, P1.1 = 0.8

Consistently a message is sent. Assuming the condition of the Markov chain that day is I, the message sent is 'acceptable" with likelihood and is - awful" with likelihood ch = 1 - = 0, 1

(a) If the interaction is in state 0 on Monday, what is the likelihood that a decent message is sent on Tuesday?

(b) If the cycle is in state 0 on Monday, what is the likelihood that a decent message is sent on Friday?

(c) In the since quite a while ago run, what extent of messages are acceptable?

(d) Let equivalent 1 if a decent message is sent on day n and let it equivalent 2 in any case. Is { Yn, n I } a Markov chain? Assuming this is the case, give its progress likelihood grid. If not, momentarily clarify why not.

66))

Leave An alone a bunch of states, and let be the leftover states. (a) What is the translation of

E ffiPij? iEA jeAc (b) What is the understanding of

E Pij? icA' jeA (c) Explain the character

E niPij = E

iA jeAc iette jEA

7Ti Pij

67))

Every day, one of n potential components is mentioned, the ith one with likelihood pi, I ?.. I,E7 = I. These components are consistently masterminded in an arranged rundown that is reexamined as follows: The component chose is moved to the front of the rundown with the general places of the multitude of different components staying unaltered. Characterize the state whenever to be the rundown requesting around then and note that there are n! potential states.

(a) Argue that the former is a Markov chain.

(b) For any state Li, .. in (which is a change of 1, 2, .. , n), let n , in) mean the restricting likelihood. All together for the state to be Li, .. in, it is fundamental for the last solicitation to be for ii, the last non-ii solicitation for, the last non-iior i2 demand for i3, etc. Henceforth, it seems natural that

701 in) = Pii Pa,

Pi; Pin-1

I Pt. , I - 131, I P.I Verify when n = 3 that the former are without a doubt the restricting probabilities.

68))

Assume that a populace comprises of a fixed number, say, m, of qualities in any age. Every quality is one of two potential hereditary sorts. On the off chance that precisely I (of the m) qualities of any age are of type 1, at that point the cutting edge will have j type 1 (and m - jtype 2) qualities with likelihood

(n\ I \j I \mj ,i)kin) m

j=0,1 i11

Allow X to signify the quantity of type 1 qualities in the nth age, and accept that X0 =/.

(a) Find E[X,J.

(b) What is the likelihood that at last every one of the qualities will be type 1?