Question

Let (N) (a) E(N3N4) (b) E(X3X4) (c) E(S3S4) 20 be a Poisson process with parameter A 2. Find the following:

Answer 1

Pg Noo Answer: - let (Ni b e a poisson process with parameler a) El N3 Ny) con that kle know that NCE) N(4+5)= NE) (NIE)+N (4+5) = N(t] E (NCE). N(H+3)]: EllN12P]+EINID IN (H5) - N(4))? El CNCtypy] + [alt]. E[NCs)] Here, we can use the property at Stationary and Independent increments. Since for tso Net) a poisson (at) where : Elnet)] = at var [n(t)} = at

pague that TE Follcas E[Net). NICZ+5)] = 9++ (atz? +asat 0 In our questions and sal, 2=2. Substitute these values in Egn o E INCB). N (341] = 2x3 + (2x3)* + 2x1x2x3 E[N3 N4} = 6+36+12 E[Ng Ny} = 54 - E[ 3 N4? = 54 b) E(X3 X4) We know that (NE){ so w poisson distribution (a). them xi did Exponential (0). ahere, ;= 1, 2, 3....

9 No Xi is ith Inter arrival time, which is Independendo '[[X3X4) = f(x3). £(X4) - trebuie - 114. Here, " f(x): Exponential distribution Ella Elxsxy)= 4] c) El 5354) (NE)t zo n poisson (2) then Snow. Gamma (n,x) [ie] n = 1,2,3.... Sn = ET

onbd ELB2,4): F{ - EALA (Žait x)} - & xjp + x4103 f$ + x4 (21+%94%)} =£(80%+ ( *184) + (2224) + f(x334) - Vay( 53) + [6(53))+ € (21) E (74) + f(x2).f(24) +6(29) f(24)] . = 4 +1+1+1 +1 9 4 4 u 3349+1+1+! For Inn Gamma (h, 2); E(Sn): A 4 and var (sn) = 1/2 LE(53,54=_3.75)