Final answer:
To calculate various probability functions from a joint PDF, one must perform integrations over the given ranges for each variable and normalize the function when calculating conditional probabilities.
Step-by-step explanation:
The task involves finding various probability functions and conditional probability functions given a joint probability density function (PDF). To find f_x(x) and f_Y(y), we must integrate the given joint PDF over the other variable's range. For f_x(x), integrate f_xy over the range of y, and for f_Y(y), integrate f_xy over the range of x. Conditional probabilities such as f_x(x|Y = 9) and f_y(y|X = 2) can be found by fixing one variable and normalizing the joint PDF over the other. The function f_x(x|Y = y) for all y is also obtained through normalization by conditioning on Y.
For example, if the joint PDF is f_xy(x, y) = Ax over the specified ranges, to find f_x(x), we would integrate from 3x to 20 with respect to y. Similarly, to find f_Y(y), we integrate from 0 to 4 with respect to x. The conditional function f_x(x|Y = y) for a specific y is the joint PDF divided by f_Y(y).