The following function is proposed as a cumulative distribution function for the bivariate random variable \((X, Y)\) :

\(F(x, y)=\left(1+e^{-(x / 10+y / 20)}-e^{-x / 10}-e^{-y / 20}ight) I_{(0, \infty)}(x) I_{(0, \infty)}(y)\)

a. Verify that the function has the appropriate properties to serve as a cumulative distribution function.

b. Derive the marginal cumulative distribution function of \(Y\).

c. Derive the marginal PDF of \(Y\).

d. Derive the joint PDF of \((X, Y)\).

e. What is the probability that \(X \leq 10\) and \(Y \leq 20\) ?

f. Are \(X\) and \(Y\) independent random variables?