Question

Let X₁ and X₂ be independent standard normal variables X₁ ∼ N(0, 1) and X₂ ∼ N(0, 1).

1) Let Y₁ = X1² + X₁² and Y₂ = X₁²− X₂² . Find the joint p.d.f. of Y₁ and Y₂, and the marginal p.d.f. of Y₁. Are Y₁ and Y₂ independent?

2) Let W = √X₁X₂/(X₁² +X₂²) . Find the p.d.f. of W.

Answer 1

Answer:-

Given That:-

Let X₁ and X₂ be independent standard normal variables X₁ ∼ N(0, 1) and X₂ ∼ N(0, 1).

1) Let Y₁ = X1² + X₁² and Y₂ = X₁²− X₂² . Find the joint p.d.f. of Y₁ and Y₂, and the marginal p.d.f. of Y₁. Are Y₁ and Y₂ independent?

Given,

independently

and

and

Jacobian:

PDF of :

  

where is the modified bessel function of the second kind are not independent.

2) Let W = √X₁X₂/(X₁² +X₂²) . Find the p.d.f. of W.

Let

and , and

Since Then,

Jacobian:

PDF of (W,Z):

PDF of W:

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