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A random process consists of three samples function
X(t,s1 ) = 2,X(t,s2) = 2 cos t1 and X(t,s3) =sin t
each occurring with equal probability. The process is;
- a)First order stationary
- b)Second order stationary
- c)Wide-sense stationary
- d)Not stationary in any sense
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
A random process consists of three samples functionX(t,s1 ) = 2,X(t,s...
Explanation:
First, let's understand the definitions of the different types of stationarity:
1. First Order Stationary: A random process is said to be first order stationary if its mean and variance are constant over time. In other words, the statistical properties of the process do not change with time.
2. Second Order Stationary: A random process is said to be second order stationary if its mean, variance, and autocovariance are constant over time. In other words, the statistical properties of the process do not change with time, and the autocovariance only depends on the time difference between two samples.
3. Wide-Sense Stationary: A random process is said to be wide-sense stationary if its mean is constant over time, and its autocovariance only depends on the time difference between two samples.
Now, let's analyze each sample of the random process given:
1. X(t, s1) = 2: This sample is a constant, which means its mean and variance are constant over time. Therefore, it is first order stationary.
2. X(t, s2) = 2 cos(t1): This sample is a cosine function with a variable phase angle t1. The mean and variance of this sample change with time due to the varying phase angle. Therefore, it is not first order stationary.
3. X(t, s3) = sin(t): This sample is a sine function with a fixed phase angle. The mean and variance of this sample are constant over time. Therefore, it is first order stationary.
Since the random process consists of three samples, and not all samples have constant mean and variance over time, the process as a whole is not first order stationary.
Furthermore, since the second sample (X(t, s2) = 2 cos(t1)) does not have a constant mean, variance, and autocovariance over time, the process is not second order stationary.
Finally, since the second sample (X(t, s2) = 2 cos(t1)) does not have a constant mean over time, the process is not wide-sense stationary.
Therefore, the correct answer is option 'D' - the process is not stationary in any sense.
First, let's understand the definitions of the different types of stationarity:
1. First Order Stationary: A random process is said to be first order stationary if its mean and variance are constant over time. In other words, the statistical properties of the process do not change with time.
2. Second Order Stationary: A random process is said to be second order stationary if its mean, variance, and autocovariance are constant over time. In other words, the statistical properties of the process do not change with time, and the autocovariance only depends on the time difference between two samples.
3. Wide-Sense Stationary: A random process is said to be wide-sense stationary if its mean is constant over time, and its autocovariance only depends on the time difference between two samples.
Now, let's analyze each sample of the random process given:
1. X(t, s1) = 2: This sample is a constant, which means its mean and variance are constant over time. Therefore, it is first order stationary.
2. X(t, s2) = 2 cos(t1): This sample is a cosine function with a variable phase angle t1. The mean and variance of this sample change with time due to the varying phase angle. Therefore, it is not first order stationary.
3. X(t, s3) = sin(t): This sample is a sine function with a fixed phase angle. The mean and variance of this sample are constant over time. Therefore, it is first order stationary.
Since the random process consists of three samples, and not all samples have constant mean and variance over time, the process as a whole is not first order stationary.
Furthermore, since the second sample (X(t, s2) = 2 cos(t1)) does not have a constant mean, variance, and autocovariance over time, the process is not second order stationary.
Finally, since the second sample (X(t, s2) = 2 cos(t1)) does not have a constant mean over time, the process is not wide-sense stationary.
Therefore, the correct answer is option 'D' - the process is not stationary in any sense.
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Community Answer
A random process consists of three samples functionX(t,s1 ) = 2,X(t,s...
Let x1 = 2,x2 = 2cost1
And x3 = 3sint
Then
fx(x) = 1/3δ(x − x1) + 1/3δ(x − x2) + 1/3δ(x − x3)
And
E[X(t)]=∫∞−∞xfx(x)dx
= 1/3[2 + 2cost + 3sint]
The mean value is time dependent so X(t) is not stationary in any sense.
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