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The random process Z(t) is defined as Z(t) = X +Y; where X and Y are independent random variables. X is uniformly distributed on (-1, 1) and Y is uniformly distributed on (0, 1)
The auto correlation function of Z(t) is Rz (0) is
- a)2/3
- b)4/3
- c)1/2
- d)1/4
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The random process Z(t) is defined as Z(t) = X +Y; where X and Y are i...
∵ X and Y are independent
Rz(0) = E[X2] + E[Y2]
Most Upvoted Answer
The random process Z(t) is defined as Z(t) = X +Y; where X and Y are i...
Understanding the Random Process Z(t)
The process Z(t) is defined as Z(t) = X + Y, where X and Y are independent random variables.
Distribution of X and Y
- X is uniformly distributed on the interval (-1, 1).
- Y is uniformly distributed on the interval (0, 1).
Calculating the Mean of X and Y
- The mean of X:
- Since X is uniformly distributed, its mean is the midpoint of the interval:
- Mean(X) = (1 + (-1))/2 = 0.
- The mean of Y:
- Mean(Y) = (1 + 0)/2 = 0.5.
Calculating the Variance of X and Y
- The variance of X:
- Variance(X) = (b - a)² / 12 = (1 - (-1))² / 12 = 4 / 12 = 1/3.
- The variance of Y:
- Variance(Y) = (1 - 0)² / 12 = 1 / 12.
Calculating the Variance of Z(t)
Since X and Y are independent, the variance of Z(t) is the sum of the variances of X and Y:
- Variance(Z) = Variance(X) + Variance(Y)
- Variance(Z) = (1/3) + (1/12)
- To add these, convert 1/3 to have a common denominator:
- Variance(Z) = (4/12) + (1/12) = 5/12.
Autocorrelation Function Rz(0)
The autocorrelation function at zero lag is given by:
- Rz(0) = E[Z(t)²] = Variance(Z) + (Mean(Z))².
- Mean(Z) = Mean(X) + Mean(Y) = 0 + 0.5 = 0.5.
Now, calculate Rz(0):
- Rz(0) = Variance(Z) + (Mean(Z))²
- Rz(0) = (5/12) + (0.5)² = (5/12) + (1/4) = (5/12) + (3/12) = 8/12 = 2/3.
Thus, the correct answer is 2/3.
The process Z(t) is defined as Z(t) = X + Y, where X and Y are independent random variables.
Distribution of X and Y
- X is uniformly distributed on the interval (-1, 1).
- Y is uniformly distributed on the interval (0, 1).
Calculating the Mean of X and Y
- The mean of X:
- Since X is uniformly distributed, its mean is the midpoint of the interval:
- Mean(X) = (1 + (-1))/2 = 0.
- The mean of Y:
- Mean(Y) = (1 + 0)/2 = 0.5.
Calculating the Variance of X and Y
- The variance of X:
- Variance(X) = (b - a)² / 12 = (1 - (-1))² / 12 = 4 / 12 = 1/3.
- The variance of Y:
- Variance(Y) = (1 - 0)² / 12 = 1 / 12.
Calculating the Variance of Z(t)
Since X and Y are independent, the variance of Z(t) is the sum of the variances of X and Y:
- Variance(Z) = Variance(X) + Variance(Y)
- Variance(Z) = (1/3) + (1/12)
- To add these, convert 1/3 to have a common denominator:
- Variance(Z) = (4/12) + (1/12) = 5/12.
Autocorrelation Function Rz(0)
The autocorrelation function at zero lag is given by:
- Rz(0) = E[Z(t)²] = Variance(Z) + (Mean(Z))².
- Mean(Z) = Mean(X) + Mean(Y) = 0 + 0.5 = 0.5.
Now, calculate Rz(0):
- Rz(0) = Variance(Z) + (Mean(Z))²
- Rz(0) = (5/12) + (0.5)² = (5/12) + (1/4) = (5/12) + (3/12) = 8/12 = 2/3.
Thus, the correct answer is 2/3.
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The random process Z(t) is defined as Z(t) = X +Y; where X and Y are independent random variables. X is uniformly distributed on (-1, 1) and Y is uniformly distributed on (0, 1)The auto correlation function of Z(t) is Rz (0) isa)2/3b)4/3c)1/2d)1/4Correct answer is option 'A'. Can you explain this answer?
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