Suppose Xn is an iid random process with Xn exponential havi

Suppose X_n is an iid random process with X_n exponential having E(X_) = 3 Find the pdf f_2, 5, 9(x_2, x_5, x_9). Find the autocovariance C(3, 11)

Solution

Xn is an iid random process with Xn exponential having E(Xn)=3

then pdf of Xn is f_n(x_n)=1/3*exp[-x_n/3]                        x_n>0

                                =0       otherwise

1. the pdf of f_2,5,9(x₂,x₅,x₉)=f₂(x₂)*f₅(x₅)*f₉(x₉)   [since Xi\'s are independent]

                                      =1/3*exp[-x₂/3] *1/3*exp[-x₅/3] *1/3*exp[-x₉/3]

                                      =1/3³*exp[-(x₂+x₅+x₉)/3]           x₂>0, x₅>0, x₉>0

2. the autocovariance C(3,11)=Cov(X₃,X₁₁)=E[X₃X₁₁]-E[X₃]*E[X₁₁]=E[X₃]*E[X₁₁]-E[X₃]*E[X₁₁]=0 [since Xi\'s are independent]

hence C(3,11)=0