(4) [40 marks] Suppose we have an election between 3 candidates A, B and C. Since all the candidates are quite bad, the voters vote randomly between the 3 candidates with probabilities $P_A$, $P_B$ and $P_C$ ($P_A + P_B + P_C = 1$). Assuming that there are $n$ independent voters and let $Y_A$ be the random variable equal to the number of votes candidate A receives.
• What is the distribution of $Y_A$? Calculate $E[Y_A]$. [5 marks]
• Let us define the joint distribution $PDF_{Y_A, Y_B, Y_C}$. For $i$, $j$, $k$ such that $i, j, k \in [0, n]$ and $i + j + k = n$, calculate $PDF_{Y_A, Y_B, Y_C}[i, j, k]$. [15 marks].
• If $P_A = 0.2$, $P_B = 0.3$ and $P_C = 0.5$ and $n = 8$, calculate the probability that A receives half the votes and candidates B and C receive a quarter of the votes each. [5 marks].
• In this same scenario, compute the probability that A receives 3 votes. [5 marks].
• In this same scenario, if A receives 3 votes, compute the probability that B and C receive 3 and 2 votes respectively. [5 marks].
• If A receives 3 votes, calculate the expected number of votes that B receives? [5 marks].