Final answer:
The PDF of W = X + Y is f_W(w) = (w^2/8), for 0 ≤ w ≤ 2 and f_W(w) = (w - 2)(4 - w)/8, for 2 < w ≤ 4.
Step-by-step explanation:
To find the probability density function (PDF) of a random variable W = X + Y, we need to integrate the joint PDF fX,Y(x, y) over the appropriate region. The joint PDF for X and Y is given by:
fX,Y(x, y) = 8xy/16, for 0 ≤ y ≤ x ≤ 2.
To find the PDF of W = X + Y, we need to find the range of values for W and then integrate fX,Y(x, y) within that range. The range of W is determined by the range of X and Y, which are both between 0 and 2. Therefore, the range of W is between 0 and 4.
To find fW(w), we need to integrate fX,Y(x, y) over the region that satisfies w = x + y. Let's consider two cases:
Case 1: 0 ≤ w ≤ 2
In this case, when w is between 0 and 2, x and y can take values such that x + y = w. The joint PDF fX,Y(x, y) will only be nonzero in this region. Since the joint PDF is constant within this region, we can calculate the area of the triangle formed by the line x + y = w and the boundaries x = 0, y = 0, and x = 2, y = 2. The area of this triangle is (w2/8).
Therefore, fW(w) = (w2/8), for 0 ≤ w ≤ 2.
Case 2: 2 < w ≤ 4
In this case, when w is between 2 and 4, x and y can take values such that x + y = w. The joint PDF fX,Y(x, y) will be nonzero in this region. Since the joint PDF is constant within this region, we can calculate the area of the trapezoid formed by the lines x + y = w, x = 0, y = 0, x = 2, and y = 2. The area of this trapezoid is (w - 2)(4 - w)/8.
Therefore, fW(w) = (w - 2)(4 - w)/8, for 2 < w ≤ 4.
In summary, the PDF of W = X + Y is:
fW(w) = (w2/8), for 0 ≤ w ≤ 2
fW(w) = (w - 2)(4 - w)/8, for 2 < w ≤ 4
0 otherwise.