Probability question

(i) Determine the Distribution of X.
Each question is an independent Bernoulli trial with probability of success 0.2. Therefore X~Binomial(n = 24, p = 0.2).

(ii) Compute P(X = 23), P(Y = 45) and P(X = 23, Y = 45). $$P(X = 23) = {24 \choose 23}*.2^{23}*.8^{1} \approx 1.6 * 10^{-15}$$ P(Y = 45) is equivalent to P(X = 23). If you guess on every question, the only way to get 45 points is if you get exactly 23 right and exactly 21 wrong. Therefore the answer is also $$1.6 * 10^{-15}$$ Similarly, the joint probability P(X = 23, Y = 45) is also $$1.6 * 10^{-15}$$

(iii) Are the random variables X and Y independent?
Clearly no. Your score is directly related to how many questions you get right.

(iv) Compute E(X) and V(X).
X is a Binomial random variable, so just use the standard formulas. $$E(X) = n*p = 24*0.2 = 4.8$$ $$V(X) = n*p*(1-p) = 24*0.2*(1-0.2)= 3.84$$

(v) Find a, b ? R such that Y = aX + b.
You score 2 points for every right answer, and -1 point for every wrong answer, so (assuming you don't leave any questions blank): $$Y = 2X - 1*(24-X)$$ $$= 2X - 24 + X$$ $$= 3X - 24$$

(vi) Compute E(Y) and V(Y).
$$E(Y) = E(3X - 24)$$ $$= E(3X) - E(24)$$ $$=3*E(X) - 24$$ $$=3*4.8 - 24$$ $$=-9.6$$ $$$$ $$V(Y) = V(3X - 24)$$ $$= V(3X) + V(24)$$ $$= 3^{2}*V(X) + 0$$ $$= 9*V(X)$$ $$= 9*3.84$$ $$= 34.56$$

(vii) Would it be better if the student gives no answer at all instead selecting purely randomly answers? Give a brief justification.
That depends on if a negative score is worse than a score of 0. If a score of 0 and a negative score are effectively the same, like if you need to reach a certain positive threshold in order to pass, then you should definitely guess on each question. Your expected score may be negative, but there will be some probability that you pass. If you don't guess on any question, your expected score will be higher (namely, 0), but the probability that you pass will be 0.

If a score of 0 is better, in some way, than a negative score, then it would be better not to guess at all, because you would be garanteed to not get a negative score.