Original practice · fully worked
Original variant: largest variance in a calibrated score family
A diagnostic score R takes values 0, 2, and 5 with probabilities q, 1/2, and 1/2-q, respectively, where 0 is at most q and q is at most 1/2. Calculate the largest possible value of Var(R).
- A
2.25 - B
2.50 - C
3.25 - D
7.00 - E
9.50
Answer and calculation
The variance is 9/4+10q-25q squared, a concave quadratic. Completing the square shows its maximum is 13/4=3.25 at q=1/5, so choice C.
Setup
Setup
Calculate the first two raw moments as functions of the probability parameter.
E[R]=2(21)+5(21−q)=27−5q E[R2]=4(21)+25(21−q)=229−25q Model
Model
Subtract the square of the mean from the second moment.
Var(R)=229−25q−(27−5q)2 =49+10q−25q2 Compute
Compute
Complete the square to expose the feasible maximizer.
Var(R)=413−25(q−51)2 q=51⟹Var(R)=413 Answer
Answer
The largest possible variance is 3.25.
3.25(C) 