Finding 6 coins in his pocket, Aloysius proposes a game of chance. He will toss all 6 coins; let X denote the number of heads obtained. Persephone will then toss all the coins which came up tails when Aloysius tossed them. Let Y denote the number of heads Persephone obtains.
1)Calculate the expectation of Y.
2)Write down the conditional probability function of X given that Y = 3.
3)Explain why you would expect the covariance between X and Y to be negative.
4)Calculate Cov(X, Y) and ρX,Y.
5)Persephone will pay Aloysius £1 for every Head Aloysius manages to obtain; Aloysius will pay Persephone £2 for every Head Persephone gets. Let Z represent Aloysius' net profit from one game. Calculate the expectation and variance of Z