1. Let Xnumpy()be randomrandom process. Let us verify the wide-sense stationarity of this process comp tation lly. Generate N realizations/sample functions of this random process (use . library), and compute (a) the mean mX(n), and (b) autocorrelation
function RX(n1, n2). Verify if the following stochastic processes are wide-sense station-ary or not, and c mpare them against theoretically obtained mean and autocorrelation. Plot the estima ed density functions for a few random variables X(k) using the func-
2. In the file Gandhinagar_RainfallData.xls, average rainfall for every month of the
tion numpy.histogram.
(a) X(n) = cos(0.2pn + q), q U[
p, p], n 2 [0, 9].
(b) X(t) = A cos(0.25pn), A U[
5, 5], n 2 [0, 7].
(c) X(n) = A( ), where A( ) N (0, 1)1 are independent random variables.
year between 1901 and 2001 is available. Treating the average rainfall every month in Gandhinagar as a Stochastic process, estimate the mean and covariance and conclude whether the stochastic process is stationary (either wide-sense or strict-sense). You might have to learn how to import data from excel file into Python for this.
3. A matrix A 2 Rn n is said to be Symmetric Positive Definite(SPD) if A = AT and yT Ay > 0, 8y 2 Rn, y 6= 0, and is said to be Symmetric Positive Semi-definite if A = AT and yT Ay 0, 8y 2 Rn. Given n random variables Xti , i = 1, . . . , n (belonging to a random process), the autocorrelation and autocovariance function of the random
vector x = (Xti )i=1,...,n can be written as a matrix R = E[xxT] and C = E[(x m)(x m)T], where m 2 Rn is the vector of means, respectively.
(a) Analytically show that the two matrices are symmetric positive semi-definite. Verify the same computationally for the two examples from the previous ques-tion2.
(b) Try to observe an additional pattern in the autocorrelation/autocovariance matrix of a wide-sense stationary process. What is such a matrix called?
1N (0, 1) denotes a normal distribution with zero mean and unit variance.
2An equivalent definition for a symmetric matrix to be positive definite (semi-definite) is that all its eigen-values are positive (non-negative).
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