(Solved): (1 point) At a certain restaurant, a group's wait time to be seated after arrival is an exponentia ...

(1 point) At a certain restaurant, a group's wait time to be seated after arrival is an exponential random variable, \( X \), with \( E[X]=20 \) minutes. Once seated, the time for the waiter to reach the table and take their order is an exponential random variable, \( Y \), with \( E[Y]=10 \) minutes. Assume \( X \) and are independent. (a) Find the joint pdf of \( X \) and \( Y \). \[ f_{X, Y}(x, y)=\left\{\begin{array}{lll} {[} & , & x>1 \\ 0 & , & \text { otherwise } \end{array}\right. \] (b) Find the cdf of \( \mathrm{W}=\mathrm{X}+\mathrm{Y} \), the total wait time. \[ F_{W}(w)=\left\{\begin{array}{ll} 1-(2 / 3) e^{\wedge}(-\mathrm{w} / 20)-(1 / 3) e^{\wedge}(-\mathrm{w} / 10) & , \\ 0 & , \text { otherwise } \end{array}\right. \] (c) Find the probability the total wait time W will be more than 30 minutes.