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Thomas Hubbard

Thomas Hubbard

Answered question

2022-05-22

I have a random variable ξ with function distribution (f.d.) G . Define the p −dimensional random vector as
X = ( ψ 1 ξ , . . . , ψ p ξ ) F ( F  is f.d. )

Given a > 0, consider A = [ a , a ] p . I want to find an expression for:
A d F
involving G and the ψ s
My attempt is first try to find the f.d. of X:
F ( x 1 , . . . , x p ) = P [ ψ 1 ξ x 1 , . . . , ψ p ξ x p ] = P [ ξ x 1 ψ 1 , . . . , ξ x p ψ p ] = P [ ξ n min { x 1 ψ 1 , . . . , x p ψ p } ] = G ( min { x 1 ψ 1 , . . . , x p ψ p } )
Even with these expressions in hand, I'm having serious trouble finding an expression for A d F.

Answer & Explanation

Travis Fernandez

Travis Fernandez

Beginner2022-05-23Added 10 answers

If I understand correctly, your task is to determine P ( X A ). First note that
P ( | ξ | a ) = P ( a ξ a ) = G ( a ) G ( a ) .
Hence, if A = [ a , a ] d , then we use arguments similar to what you wrote and the fact that ψ i are constant to get
P ( X A ) = P ( | ψ 1 ξ | a , , | ψ p ξ | a ) = P ( | ξ | a max ( | ψ 1 | , , | ψ p | ) ) = G ( a max ( | ψ 1 | , , | ψ p | ) ) G ( a max ( | ψ 1 | , , | ψ p | ) ) .

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