A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained. Following is a record of the number of students in a class of 20 who took the test each number of times.

Students Number of tests
10 1
7 2
2 3
1 4

a. Use the relative frequency approach to construct a probability distribution and show that it satisfies the required condition.

b. Find the expected value of the number of tests taken.

c. Compute the variance.

d. Compute the standard deviation.