A zero-mean Gaussian random process has an auto-correlation function

R_XX (τ)={■(13[1-(|τ|⁄6)] |τ|≤6@0 elsewhere)┤
Find the covariance function necessary to specify the joint density of random variables defined at times t_i=2(i-1),i=1,2,…,5. Give the covariance matrix for the X_i=X(t_i ).

To find the covariance function necessary to specify the joint density of random variables defined at times t_i = 2(i-1), i = 1, 2, ..., 5, we can use the relationship between the auto-correlation function and the covariance function.

The covariance function is defined as the auto-correlation function subtracted by the product of the mean squared. Given that the random process has a zero mean, we can directly use the auto-correlation function to find the covariance function.

The given auto-correlation function R_XX (τ) is:
R_XX (τ) = 13[1 - (|τ|/6)] for |τ| ≤ 6, 0 elsewhere

Since the values of τ are discrete for the given times t_i = 2(i-1), i = 1, 2, ..., 5, we need to evaluate the auto-correlation function at those specific values.

For i = 1, we have t_1 = 2(1-1) = 0. The auto-correlation function at τ = 0 is given by:
R_XX (0) = 13[1 - (|0|/6)] = 13

For i = 2, we have t_2 = 2(2-1) = 2. The auto-correlation function at τ = 2 is given by:
R_XX (2) = 13[1 - (|2|/6)] = 9.67

Similarly, for i = 3, 4, and 5, we have t_3 = 4, t_4 = 6, and t_5 = 8 respectively, and the corresponding auto-correlation function values are:
R_XX (4) = 4.33
R_XX (6) = 0
R_XX (8) = 0

Now we can form the covariance matrix for the X_i = X(t_i) random variables. The covariance matrix is given by:
C = [Cov(X_1, X_1) Cov(X_1, X_2) Cov(X_1, X_3) Cov(X_1, X_4) Cov(X_1, X_5)
Cov(X_2, X_1) Cov(X_2, X_2) Cov(X_2, X_3) Cov(X_2, X_4) Cov(X_2, X_5)
Cov(X_3, X_1) Cov(X_3, X_2) Cov(X_3, X_3) Cov(X_3, X_4) Cov(X_3, X_5)
Cov(X_4, X_1) Cov(X_4, X_2) Cov(X_4, X_3) Cov(X_4, X_4) Cov(X_4, X_5)
Cov(X_5, X_1) Cov(X_5, X_2) Cov(X_5, X_3) Cov(X_5, X_4) Cov(X_5, X_5)]

Using the information we calculated earlier, the covariance matrix becomes:
C = [13 9.67 0 0 0
9.67 13 9.67 0 0
0 9.67 13 9.67 0
0 0 9.67 13 9.67
0 0 0 9.67 13]

So, the covariance matrix for the X_i = X(t_i) random variables is:
C = [13 9.67 0 0 0
9.67 13 9.67 0 0
0 9.67 13 9.67 0
0 0 9.67 13 9.67
0 0 0 9.67 13]