Based on years of experience, an economics professor knows that on the first principles of economics exam of the semester \(13 \%\) of students will receive an A, \(22 \%\) will receive a \(\mathrm{B}, 35 \%\) will receive a C, \(20 \%\) will receive a \(D\), and the remainder will earn an \(F\). Assume a 4 point grading scale \((A=4, B=3\), \(\mathrm{C}=2, \mathrm{D}=1\), and \(\mathrm{F}=0\) ). Define the random variable \(G R A D E=4,3,2,1,0\) to be the grade of a randomly chosen student.

a. What is the probability distribution \(f(G R A D E)\) for this random variable?

b. What is the expected value of GRADE? What is the variance of GRADE? Show your work.

c. The professor has 300 students in each class. Suppose that the grade of the \(i\) th student is GRADE and that the probability distribution of grades \(f\left(G R A D E_{i}\right)\) is the same for all students. Define \(C L A S S_{-} A V G=\sum_{i=1}^{300} G R A D E_{i} / 300\). Find the expected value and variance of CLASS_AVG.

d. The professor has estimated that the number of economics majors coming from the class is related to the grade on the first exam. He believes the relationship to be MAJORS \(=50+10 C L A S S \_A V G\). Find the expected value and variance of MAJORS. Show your work.