"Q11a) Density function of T-? ? F(t) =d/dt * P[T=t) =µ?at 0=t<88 8 -? ? E(T) = ? ? ? ? ?= ? ? ? ? ?= 1/? 0 0 Average value of T is 1/?Then µ = 1/mean time unit transition = 1/(1/µ)0 -? ? P*T>t+ = *(µ)? ]/0!-? ? = ?b) EquivalenceP(T>s + t, T > s]/P(x > t) = P(x > s)P(x > s + t) = P(T > s)P(T > t)1 - ? ? P(T > t) = ?P(T > s + t) = P(T > s)P(T>µ)1 1 - ?- ? ? ?= ? * ?1 - (? +? ) ?= ?c) P(A|B) = P(A n? )/? (? )letT be the variable representing the inter- arrival time between the successive arrivals at two timepoints-? ? P(T= ? ) = 1- ?-? ? No arrival yet for a period of t second P(T= ? ) = ?Prove that P(T= ? + ?? |? = ? ) = P(0 = ? = ?? )P(T= ? + ?? ? = ? = ? [ ? = ? + ?? n ? = ? ]/? (? = ? )-? ? -? ? + ?? -? ? -? (?? ) = ? - ? ]/?= 1- ?= P(0 = ? = ?? )1 Q12) P (t) = ?*P (t) – P (t)]n n+1 n n=01 P (t) = -?P (t)0 0 Showing by induction the system of differential equation has solution given by the poisson process-? ? ? P (t) = ? ? ? /? !For t = 0,? = 0,1,2…n SolnLet consider time dependent function be N(t) and denotes the number of arrivals that occur in agiven interval [0,t]Probability arrival occurs in a short interval *t, t + ?t+ is proportion to the length of interval ?tP*N(t + ?t) – N(t) = 1] = ??t for ?>0 Probability that arrival occurs in *t, t + ?t+ does not depend on the time of previous arrivals P*N(t +?t) – N(t) =n |N(s) = m+ = P*N(t + ?t) – N(t) = n] for all 0= ? = ? ? ? ? ? ? ? h? ? ? ? ? ? ? ? ? ? ? ? ? ? ?*t, t + ?t+ is negligibleIf ?t = 0 then P (t) = P[N(t) = n]n N(t) has possible values 0,1,2…Consider a time step of size?t then N(t) . The transition matrix for chain1- ? ? ? 0 0 0 …? ? ? 1- ? ? ?0 0 … A = .. . .… X = AXt + ?t t P (t +?t) = (1 – ??t)P (t)0 0 P (t + ?t) = ??tP (t) + (1 – ??t)P (t)1 0 1 1 P = lim ? (t + ?t) – P (t)/?tn ? ? >0 n n= ?*P (t) – P (t)n-1 n Conditions: no arrival or P (0) =10 1 n = 1 ,P (t) = ?*P (t) – P (t)]t 0 1 -? ? = ?*? - P (t)]1 First order differential equation? ? d/dt(? P (t) = ?1 ? ? ? P (t) = ?t + c1 Where C is arbitrary constantWhere P (0) 0 implies P (0) = 0 for n= 10 n-? ? P (t) = ?t?1 System differential equation is therefore -? ? ?P (t) = ? ? ? /? !For t = 0,? = 0,1,2…n Q13) The system differential equation 1 P (t) = -?P (t) + µP (t)0 0 t 1 P = lim ? (t + ?t) – P (t)/?tn n n ? ? >0= ?P (t) – (? + µ)P (t) + µP (t) for n = 1n-1 n n+1 1 P (t) =0 for = 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?n Assuming 1 P = lim ? (t) = P where P does not depend upon tn n n 1 ? ? >8 Therefore-?P + µP = 00 1 1 ?P – (? + µ)P + µP for n= 1n-1 n n +1 solvingP = (?/µ)P1 0 n= 1?P – (? + µ)P + µP = 00 0 2 so 2 P = (?/µ) P 2 0Induction argumentn P = (?/µ) Pn 0 P + P + P + …0 1 2 Thus ? ? 8 P = 1 00 ? Q14a)1 P (t) = ?*P (t) – P (t)]n n-1 n 1 P (t) = ?P (t)0 0 Conditions: no arrival or P (0) =10 -? ? P (t) = ?0 1 n = 1 ,P (t) = ?*P (t) – P (t)]t 0 1 -? ? = ?*? - P (t)]1 ? ? First order differential equation multiplying ?? ? d/dt(? P (t) = ?1 ? ? ? P (t) = ?t + c1 Where C is arbitrary constantHence= ?t b) since ?= average number of occurrence, per unit timet= time-dependent distribution for number of occurrence ?t = the number of customers on average number of occurrence per unit = mean of Poisson process= ?t "