Question

let y1 and y2 have joint density function f(y1, y2) and marginal densities f1(y1) and f2(y2), respectively. show that y1 and y2 are independent if and only if f(y1|y2)

Answer 1

To show that y1 and y2 are independent, we need to prove that their joint density function f(y1,y2) is equal to the product of their marginal densities f1(y1) and f2(y2).

Step-by-step explanation:

To show that y1 and y2 are independent, we need to prove that their joint density function f(y1,y2) is equal to the product of their marginal densities f1(y1) and f2(y2). In other words, f(y1,y2) = f1(y1) * f2(y2).

To do this, we can start by using the definition of conditional probability and the fact that two events A and B are independent if and only if P(A|B) = P(A).

So, if y1 and y2 are independent, then P(y1|y2) = P(y1). This means that the conditional density function f(y1|y2) is equal to the marginal density function f1(y1), since the conditional probability P(y1|y2) is equal to the marginal probability P(y1). Therefore, f(y1|y2) = f1(y1), which implies that f(y1,y2) = f1(y1) * f2(y2).