a Find PX10 8 X3 6 and EX10X3 Let Xtt or to 0 be a Poiss

(a) Find P(X₁₀ = 8, X₃ = 6) and E(X₁₀X₃)

Let (Xt)t > or = to 0 be a Poisson process with intensity 3 and let S0,S1,S2,... be its interarrival (or sojourn) times. (a) Find P(X10 = 8, X3 = 6) and E(X10X3)

Solution

When X is a Poisson process with intensity 3, the variables S0,S1, S2.... follow an exponential process with parameter 1/3

a) P(X10 =8, x3 =6) = P(x10=8) * P(X3 =6) (as xis are independent)

= 0.0081(0.0504)

= 0.000040824

= 0.00041

E(x10x3) = E(x10)E(x3) = 3*3 = 9