Show that if X 1 , X 2 ,, X p are independent, continuous random variables, P(X 1 A 1 , X 2 A 2 ,, X p A p ) P(X 1 A 1 )P(X 2 A 2 ) P(X p A p ) for any regions A 1 , A 2 ,, A p in the range of X 1 , X 2 ,, X p respectively ...

Show that if X 1 , X 2 ,, X p are independent, continuous random variables, P(X

Question:

Show that if X₁, X₂,…, X_p are independent, continuous random variables, P(X₁ ∈ A₁, X₂ ∈ A₂,…, X_p ∈ A_p) = P(X₁ ∈ A₁)P(X₂∈ A₂) … P(X_p ∈ A_p) for any regions A₁, A₂,…, A_p in the range of X₁, X₂,…, X_p respectively.