(a) The joint cumulative distribution function (CDF) of X and Y is obtained by integrating the joint density function over the range of the variables. The CDF, F(x,y), is given by:

F(x,y) = ∫∫f(u,v) du dv, where the double integral is over the region 0 ≤ u ≤ x and 0 ≤ v ≤ y.

So, we have:

F(x,y) = ∫ (from 0 to x) ∫ (from 0 to y) (2u + v)/20 du dv = ∫ (from 0 to x) [(u^2 + uv)/20] (from 0 to y) du = ∫ (from 0 to x) [y^2/20 + xy/20] du = [(x^2y/20 + xy^2/40)] (from 0 to x) = x^2y/20 + xy^2/40

(b) X and Y are independent if and only if the joint density function f(x,y) can be written as the product of the marginal density functions of X and Y.

The marginal density functions are obtained by integrating the joint density function over the range of the other variable.

For X, we have:

f_X(x) = ∫ (from 0 to 2) f(x,y) dy = ∫ (from 0 to 2) (2x + y)/20 dy = [2xy/20 + y^2/40] (from 0 to 2) = 2x/10 + 1/5 = x/5 + 1/5

For Y, we have:

f_Y(y) = ∫ (from 0 to 2) f(x,y) dx = ∫ (from 0 to 2) (2x + y)/20 dx = [x^2/20 + xy/20] (from 0 to 2) = 2/20 + y/10 = 1/10 + y/10

The product of the marginal densities is:

f_X(x) * f_Y(y) = (x/5 + 1/5) * (1/10 + y/10) = x/50 + y/50 + x*y/50 + y/100

This is not equal to the joint density function f(x,y) = (2x + y)/20, so X and Y are not independent.