Suppose Z(t)=sum_{k=1}^{n}Xe^{j(omega_{0}t+Phi_{k})}, t in R where omega_{0} is a constant...

Mbalisikerc 2022-07-16 Answered
Suppose Z ( t ) = Ξ£ k = 1 n X e j ( πœ” 0 t + 𝚽 k ) , t ∈ R where πœ” 0 is a constant, n is a fixed positive integer, X 1 , . . . , X n , γ€€ 𝚽 1 , . . . , 𝚽 n are mutually independent random variables, and E X k = 0 , D X k = Οƒ k 2 , 𝚽 , U [ 0 , 2 Ο€ ] , k = 1 , 2 , . . . , n . Find the mean function and correlation function of { Z ( t ) , Β  t ∈ R } .
I have tried to solve it.
For mean function,
m Z ( s ) = E { Z s } = E { X s } + i E { Y t }
= E { Ξ£ k = 1 s X e j ( πœ” 0 t + 𝚽 k ) }
For correlation function,
R Z ( s , u ) = E { Z s , Z u }
= E { Y ( s ) Y ( u ) }
= E { Ξ£ k = 1 s X e j ( πœ” 0 t + 𝚽 k ) Ξ£ k = 1 u X e j ( πœ” 0 t + 𝚽 k ) }
= E { Ξ£ k = 1 s Ξ£ k = 1 u X e j ( πœ” 0 t + 𝚽 k ) X e j ( πœ” 0 t + 𝚽 k ) }
I am stuck here. How to move from here ahead?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

suponeriq
Answered 2022-07-17 Author has 10 answers
Step 1
Before finding the mean and variance function, one has to fix the notation of Z ( t ), which should be
Z ( t ) = βˆ‘ k = 1 n X k e j ( πœ” 0 t + Ξ¦ k ) . .
In order to compute the mean of Z ( t ), it suffices to compute the expectation of X k e j ( πœ” 0 t + Ξ¦ k ) . To do so, use the independence between X k and Ξ¦ k and the fact that X k is centered.
For the correlation function, the most technical part is to compute Cov ⁑ ( Z ( s ) , Z ( t ) ) which actually reduces to
βˆ‘ k = 1 n βˆ‘ β„“ = 1 n E [ X k X β„“ e j ( Ο‰ 0 s + Ξ¦ k ) e j ( Ο‰ 0 t + Ξ¦ β„“ ) ]
(when you have to sums, it is always preferable to sum over different indices). If k β‰  β„“ , the corresponding term vanishes.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-19
Contribution of each variable in multiple linear regression
What will be the best measure of the contribution of a variable in multiple linear regression? I was thinking of using the coefficient ratio as a marker of a variable's contribution.
For example:
If the equation is
Y p r e d i c t e d = a 1 X 1 + a 2 X 2 + a 3
Then X 1 's contribution can be written down as:
a 1 a 1 + a 2 + a 3
Is there some other method possible to write down the contribution. Since in this case if any coefficient is negative, there is a possibility that a variable's contribution exceeds 100 %
asked 2022-07-14
Why is it valid to use squared Euclidean distances in high dimensions in multiple regression?
Euclidean distance is not linear in high dimensions. However, in multiple regression the idea is to minimize square distances from data points to a hyperplane.
Other data analysis techniques have been considered problematic for their reliance on Euclidean distances (nearest neighbors), and dimensionality reduction techniques have been proposed.
Why is this not a problem in multiple regression?
asked 2022-05-08
In Linear regression, we have ΞΈ = ( X T X ) βˆ’ 1 X T y .
In Ridge regression, we have ΞΈ = ( Ξ» I + X T X ) βˆ’ 1 X T y .
I learnt somewhere that while X T X is not guaranteed to be invertible, Ξ» I + X T X is guaranteed to be invertible.
Is this true? If so, why?
asked 2022-05-11

According to government data, the probability that an adull was never in a museum is 15%. In a random survey of 10 adults, what is the probability that at least eight were in a museum? Round to three decimal places

asked 2022-06-30
If the coefficients of a simple regression line, B 0 and B 1 , are the same then why are the regression lines of y on x and x on y different given the condition r 2 < 1. I have tried all the manipulation and graphical analysis I can but can't seem to see why this is happening.
asked 2022-06-02
If the joint density function of X and Y is given by:
f ( x , y ) = { 1 / 2 , forΒ  | x | + | y | ≀ 1 0 , otherwise
Show that Y has constant regression with respect to X and/but that X and Y are not independant.
asked 2022-07-16
PCA vs Correlation
What is the relationship between (first) principal component(s) and the correlation matrix or the average correlation of the data. For example, in an empirical application I observe that the average correlation is almost the same as the ratio of the variance of the first principal component (first eigenvalue) to the total variance (sum of all eigenvalues).
Is there a mathematical relationship?

New questions

The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual racetracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The pro?t contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:
Max 200DRB + 280DRW
s.t.
20DRB + 25DRW 40,000 Steel Available
40DRB + 100DRW ? 120,000 Manufacturing minutes
60DRB + 40DRW ? 96,000 Assembly minutes
DRB, DRW ? 0
Optimal Objective Value = 424000.00000
Variable Value blackuced Cost
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 1000.00000 0.00000
DRW 800.00000 0.00000
Constraint Slack/ Surplus Dual Value
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 0.00000 8.80000
2 0.00000 0.60000
3 4000.00000 0.00000
Objective Allowable Allowable
Variable Coef?cient Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 200.00000 24.00000 88.00000
DRW 280.00000 220.00000 30.00000
RHS Allowable Allowable
Constraint Value Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 40000.00000 909.09091 10000.00000
2 120000.00000 40000.00000 5714.28571
3 96000.00000 Infnite 4000.00000
a. What are the optimal solution and the total profit contribution?
b. Another supplier offeblack to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.
c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.
d. Because of increased competition, Deegan is considering blackucing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.
e. If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.