Q 1 Let Nt be a rate A Poisson process Find PNt1 x1 Nt2 x2

Q 1 Let N(t) be a rate A Poisson process. Find P[N(t1) = x1, N(t2) = x2,N(t3) = X3] for 0

Solution

p(N(t1)=x1,N(t2=x2),N(t3)=x3)=p(N(t1)=x1)*p(N(t2)=x2/N(t1)=x1)*p(N(t3)=x3/N(t2)=x2,N(t1)=x1)

P(x; ?) = (e^-?) (?^x) / x!

we get

= (e^-?) (?^x1*t1) / (x₁*t₁)!* (e^-?) (?^x2*t2) / (x₂*t₂)!*(e^-?) (?^x3*t3) / (x₃*t₃)!