5-57. (Bayes’ Rule for continuous random variables) We may posit the multiplication theorem for probability density functions as f (x, y) = g(x)h(y|x) =

h(y)g(x|y). Then using this expression, along with (5.22), demonstrate that

(5.23) can be written as g(x|y) = g(x)h(y|x)

4

+∞

−∞ g(x)h(y|x)dx

.

Similarly, using the preceding multiplication theorem along with (5.21), verify that (5.24) can be expressed as h(y|x) = h(y)g(x|y)

4

+∞

−∞ h(y)g(x|y)dy

.

Given the probability density function f (x, y) = -x + y, 0< x < 1 and 0 < y < 1;

0 elsewhere, use Bayes’ Rule to find g(x|y) and h(y|x).