3
$\begingroup$

If R.V. A is independent of a random vector (B,C), is A necessarily independent of C?

  • 5
    Write down the definition of independence, and you will successfully formalize a result quite intuitively obvious (if you are independent of what happens to $(B,C)$, you are independent of what happens to $C$).2012-10-22
  • 0
    By the way, welcome to math.stackexchange.2012-10-22
  • 0
    Thank you, I integrated both sides of the definition over variable B to get the result.2012-10-22
  • 2
    You can answer your own question.2012-10-22

2 Answers 2

1

By independence of a and vector (b,c), we have $$f(a,b,c)=f(a)f(b,c)$$ So, integrating out b, $$\int_{-\infty}^{\infty} f(a,b,c)\,db = \int_{-\infty}^{\infty} f(a)f(b,c)\,db \implies f(a,c)=f(a)f(c) $$

0

If you go back to the definition of independence, you can prove this statement without assuming the joint distribution of $(A,B,C)$ to be continuous.

$A$ and $(B,C)$ are independent if every $A$-measurable event is independent of every $(B,C)$-measurable event. Since every $C$-measurable event is $(B,C)$-measurable, conclude that every $A$-measurable event is independent of every $C$-measurable event. That is, $A$ and $C$ are independent.