Final answer:
To find the joint PDF and marginal density functions, analyze the given region on the Cartesian plane and calculate the probabilities. The joint PDF is 1/4 and the marginal density functions can be found by integrating the joint PDF. To determine independence, check if the joint PDF can be factored into the marginal density functions.
Step-by-step explanation:
To find the joint PDF (probability density function) and the marginal density functions, we need to understand the given region on the Cartesian plane. The region is defined by the inequality |x| + 2|y| ≤ 2. This represents a rectangle with vertices (-2,0), (0,1), (2,0), and (0,-1). Since the point is chosen at random, the probability is uniformly distributed over this region.
The joint PDF fXY(x,y) is equal to 1 divided by the total area of the region. In this case, the area of the region is 4, so the joint PDF is 1/4.
The marginal density functions fX(x) and fY(y) can be found by integrating the joint PDF over the range of the variable. For example, fX(x) is obtained by integrating fXY(x,y) with respect to y, from -1 to 1. Similarly, fY(y) is obtained by integrating fXY(x,y) with respect to x, from -2 to 2.
To determine if X and Y are independent random variables, we need to check if fXY(x,y) can be factored into fX(x) and fY(y). If it can be factored, then X and Y are independent. If not, then they are dependent.