Let X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX(x) and fY(y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative.

a. Show that there exists a positive constant c such that fX(x) = cg(x) and fY(y) = (1/c)h(y).

b. Use part (a) to show that X and Y are independent.

Answer

Step 1 of 3</p>

a) Given probability density function

is a function of X alone

is a function of Y alone

= c

When you differentiating the joint pdf with respect to y

We get function of ‘X’ with some constant c

Step 2 of 3</p>

= (1/c)

When you differentiating the joint pdf with respect to X

We get function of ‘y’ with some constant 1/c