Question on time series/cauchy distribution

[caspian2012]

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!

 

[caspian2012]

This is my first post so I guess I will try to type everything again. Hopefully it works this time.

 

[caspian2012]

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 ?]

 

[Ray Vickson]

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!

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

 

[caspian2012]

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

Thanks. I thought it would be much more tricky, guess not.