In a test, each question has 4 options of which exactly one is correct. A correct answer gets 4 marks and a wrong answer gets −1 mark. A student answers a question by pure guessing, and X denotes the marks he gets for it. Which of the following are correct?
(A) \(\displaystyle E(X) = \frac{1}{4}\)
(B) \(\displaystyle \mathrm{Var}(X) = \frac{19}{4}\)
(C) If he guesses 40 questions in this way, his expected total marks are 10.
(D) If a wrong answer got −2 marks instead, the expected marks for a guessed question would be 0.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
X takes the value 4 with probability \(\displaystyle\frac{1}{4}\) and −1 with probability \(\displaystyle\frac{3}{4}\).
\( \displaystyle E(X) = 4 \times \frac{1}{4} + (-1) \times \frac{3}{4} = \frac{1}{4} \)
\( \displaystyle E(X^{2}) = 16 \times \frac{1}{4} + 1 \times \frac{3}{4} = \frac{19}{4} \)
\( \displaystyle \mathrm{Var}(X) = \frac{19}{4} - \frac{1}{16} = \frac{75}{16} \)
So \(E(X) = \frac{1}{4}\), \(E(X^{2}) = \frac{19}{4}\) and \(\mathrm{Var}(X) = \frac{75}{16}\); expected marks add over questions.
Checking the statements
(A) \(\displaystyle E(X) = \frac{1}{4}\) — Correct. \(1 - \frac{3}{4} = \frac{1}{4}\).
(B) \(\displaystyle \mathrm{Var}(X) = \frac{19}{4}\) — Incorrect. \(\frac{19}{4}\) is \(E(X^{2})\); the variance is \(\frac{75}{16}\).
(C) If he guesses 40 questions in this way, his expected total marks are 10 — Correct. \(40 \times \frac{1}{4} = 10\).
(D) With −2 for a wrong answer, the expected marks for a guess would be 0 — Incorrect. \(4 \times \frac{1}{4} - 2 \times \frac{3}{4} = -\frac{1}{2}\).