Problem 1 [10 marks]

A game consists of first rolling a fair 10-faced die once and then tossing a coin 10 times. The number rolled on the die will determine the probability of landing heads for the coin. Let R be the random variable for the value rolled from the die, and N be the random variable for the number of heads from the 10 coin tosses. If R = r is rolled, the coin has probability r/10 of landing heads up.

(a) What is the conditional expectation of N given R = r? [2.5 marks]

(b) What is the conditional variance of N given R = r? [2.5 marks]

(c) Use your results from parts (a) and (b) to compute the expected value of the number of heads obtained in playing the game once. [2.5 marks]

(d) Use your results from parts (a) and (b) to compute the variance of the number of heads obtained in playing the game once. [2.5 marks]

Problem 1 [10 marks] A game consists of rst rolling a fair 10-faced die once and then tossing a coin 10 times. The number rolled on the die will determine the probability of landing heads for the coin. Let R be the random variable for the value rolled from the die, and N be the random variable for the number of heads from the 10 coin tosses. If R = r is rolled, the coin has probability r/ 10 of landing heads up. (a) What is the conditional expectation of N given R = r? [2.5 marks] (b) What is the conditional variance of N given R = r? [2.5 marks] (c) Use your results from parts (a) and (b) to compute the expected value of the number of heads obtained in playing the game once. [2.5 marks] (d) Use your results from parts (a) and (b) to compute the variance of the number of heads obtained in playing the game once. [2.5 marks]