A student must attempt all three problems that appear on a c

A student must attempt all three problems that appear on a chemistry exam. The probability that the student will solve the first problem correctly is 0.8, the second 0.3, and the third 0.5. Let X denote the number of problems that are solved correctly by the student.

a. Find the probability mass function (PMF) of X.

b. What is the average number of problems that the student can solve correctly?

c. What is the variance of X?

Solution

a. P (X=x) = {x=0, 0.2*0.7*0.5 = 0.07;

x=1, 0.8*0.7*0.5 + 0.2*0.3*0.5 + 0.2*0.7*0.5 = 0.38

x=2, 0.8*0.3*0.5 + 0.8*0.7*0.5 + 0.2*0.3*0.5 = 0.43

x=3, 0.8*0.3*0.5 = 0.12}

b. E(X) = 0.07*0 + 0.38*1 + 0.43*2 + 0.12*3 = 1.6

c. E(X^2) = 0.07*0^2 + 0.38*1^2 + 0.43*2^2 + 0.12*3^2 = 3.18

Var (X) = E(X^2) - E(X)^2 = 3.18 - 2.56 = 0.62