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Question on time series/cauchy distribution

  1. Oct 4, 2012 #1
    1. (Form 1) X_t = A cos(λt +φ ) ,
    where φ was Unif [−pi ,pi ] , λ is a fixed constant and A is a constant (or a RV mean 0,
    variance \σ^\2_\A, and indep of φ ). ]
    Now consider the following: Let B_1 , B_2 be IID Normal(0,\σ^\2_\B) and λ a fixed constant
    (Form 2) Y_t = B_1 cos(λ_t) + B2 sin(λ_t)
    Can Form 2 be written in a manner similar to Form 1? If so, show how.
    [Hint: you know the distribution of {\frac{\-B_2}{B_1}}
    (Cauchy). Define
    φ = arctan({\frac{\-B_2}{B_1}}) . What is the distribution of φ ?]


    I am lost in what I need to do. Could someone give me some help? Or some hint?

    Thanks!
     
  2. jcsd
  3. Oct 4, 2012 #2
    This is my first post so I guess I will try to type everything again. Hopefully it works this time.
     
  4. Oct 4, 2012 #3
    1. (Form 1) [tex]X_t [/tex]= A cos(λt +φ ) ,
    where φ was Unif [−pi ,pi ] , λ is a fixed constant and A is a constant (or a RV mean 0,
    variance [tex]\σ^\2_\A[/tex], and indep of φ ). ]
    Now consider the following: Let [tex]B_1[/tex] , [tex]B_2 [/tex]be IID Normal(0,[tex]\σ^\2_\B[/tex]) and λ a fixed constant
    (Form 2) [tex]Y_t[/tex] = [tex]B_1[/tex] cos([tex]λ_t[/tex]) + B2 sin([tex]λ_t[/tex])
    Can Form 2 be written in a manner similar to Form 1? If so, show how.
    [Hint: you know the distribution of [tex]{\frac{\-B_2}{B_1}}[/tex]
    (Cauchy). Define
    φ = arctan([tex]{\frac{\-B_2}{B_1}}[/tex]) . What is the distribution of φ ?]
     
  5. Oct 4, 2012 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Just use standard trigonometric identities; in particular, how can you write sin(a+b) in terms of sin(a), sin(b), cos(a) and cos(b)?

    RGV
     
  6. Oct 4, 2012 #5
    Thanks. I thought it would be much more tricky, guess not.
     
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