A random process consists of three samples function X(t, s1 ) = 2, X(t, s2 ) = 2cos t1 and X(t, s3 ) = 3sint - each occurring with equal probability. The process isa)First order stationaryb)Second order stationaryc)Wide-sense stationaryd)Not stationary in any senseCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation:

Random process is a collection of random variables. In this case, we have three random variables X(t,s1), X(t,s2) and X(t,s3).

First-order stationary process:
A process is said to be first-order stationary if its mean and variance are constant over time. But, in this case, the mean and variance of each random variable are different. Hence, the process is not first-order stationary.

Second-order stationary process:
A process is said to be second-order stationary if its mean, variance and autocorrelation function are constant over time. To check whether the process is second-order stationary or not, we need to calculate the autocorrelation function (ACF) of the process.

ACF of X(t,s1):
R(s,s) = E[X(t,s1)X(t+s,s1)] = 4
R(s,s+τ) = E[X(t,s1)X(t+s+τ,s1)] = 2

ACF of X(t,s2):
R(s,s) = E[X(t,s2)X(t+s,s2)] = 2
R(s,s+τ) = E[X(t,s2)X(t+s+τ,s2)] = cos τ

ACF of X(t,s3):
R(s,s) = E[X(t,s3)X(t+s,s3)] = 9/2
R(s,s+τ) = E[X(t,s3)X(t+s+τ,s3)] = 3/2 sin τ

As we can see, the ACF of each random variable is different. Hence, the process is not second-order stationary.

Wide-sense stationary process:
A process is said to be wide-sense stationary if its mean is constant over time and its autocorrelation function depends only on the time difference τ. As we can see from the ACF calculations, the ACF of each random variable depends on both s and τ. Hence, the process is not wide-sense stationary.

Conclusion:
Since the process does not satisfy the conditions of any of the stationary processes, it is not stationary in any sense.

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