In the Advanced Placement (AP) programs of high schools, during the

senior year 40% of students study subject D, 45% study subject E, and

50% study subject F. It is also known that 10% study both subjects

D and F, 20% study both subjects E and F, 15% study subject D

but neither subject E nor F. Only 5% of the students study all three

subjects.

(a) Define the random variable X as the number subjects taken (out

of D, E, and F) by AP students during the senior year. Obtain the

probability mass function and cumulative distribution function of

X and represent it in the tabular form.

(b) Obtain the expected value of X.