Question

You receive an A 40% of the time, a B 20% of the time, a C 15% of the time , and a D the rest of the time. For any random test, what is your expected grade point average? (Use the following values for X: A=4, B=3, C=2, D=1)

Answer 1

The grade you receive on a test is a discrete random variable with probability mass function

`f_X(x)=P(X=x)=\begin{cases}0.4&\text{for }X=4\text{ (i.e. scoring an A)}\\0.2&\text{for }X=3\text{ (B)}\\0.15&\text{for }X=2\text{ (C)}\\0.25&\text{for }X=1\text{ (D)}\\0&\text{otherwise}\end{cases}`

where the probability of getting a D is 25% because there are only 4 possible grades, and the probabilities must sum to 1 :

0.4 + 0.2 + 0.15 + 0.25 = 1

The expected test score/GPA is then

`\displaystyle\sum_x x\,f_X(x)=\sum_(x=1)^4 x\,P(X=x)=1\cdot0.25+2\cdot0.15+3\cdot0.2+4\cdot0.4=\boxed{2.75}`

so your average would fall between a B and a C.