Suppose you are playing a simple game that costs you $5 per play, where you draw one cardfrom a well-shuffled deck of 52 cards.• If the card is a spade, you receive $20. (You don’t get the $5 back.)• If it is not, then you do not receive any money. (You don’t get the $5 back.)Let X = the amount of money you have before playing a round of this “game”.Let Y = the amount of money you have after playing a round of this “game”.(a) Compute the conditional expectation of Y given X = $100.(b) You can also use the following variant of the Law of Total Expectation tocompute a conditional expectation:E(Z | X = x) = Summation(E(Z | Ai ∩ X = x)P (Ai | X = x)For this part, let Z = the amount of money you have after playing TWO rounds ofthis “game”. Use the above formula to calculate the conditional expectation of Z givenX = $100. (Note: the cards are well-shuffled again between rounds.)(c) Now suppose you get only $16 for drawing a spade. Compute the conditionalexpectation of Y (not Z) given X = $100
Suppose you are playing a simple game that costs you $5 per play, where you draw one cardfrom a well-shuffled deck of 52 cards.• If the card is a spade, you receive $20. (You don’t get the $5 back.)• If it is not, then you do not receive any money. (You don’t get the $5 back.)Let X = the amount of money you have before playing a round of this “game”.Let Y = the amount of money you have after playing a round of this “game”.(a) Compute the conditional expectation of Y given X = $100.(b) You can also use the following variant of the Law of Total Expectation tocompute a conditional expectation:E(Z | X = x) = Summation(E(Z | Ai ∩ X = x)P (Ai | X = x)For this part, let Z = the amount of money you have after playing TWO rounds ofthis “game”. Use the above formula to calculate the conditional expectation of Z givenX = $100. (Note: the cards are well-shuffled again between rounds.)(c) Now suppose you get only $16 for drawing a spade. Compute the conditionalexpectation of Y (not Z) given X = $100
Suppose you are playing a simple game that costs you $5 per play, where you draw one card from a well-shuffled deck of 52 cards. • If the card is a spade, you receive $20. (You don’t get the $5 back.) • If it is not, then you do not receive any money. (You don’t get the $5 back.) Let X = the amount of money you have before playing a round of this “game”. Let Y = the amount of money you have after playing a round of this “game”. (a) Compute the conditional expectation of Y given X = $100. (b) You can also use the following variant of the Law of Total Expectation to compute a conditional expectation: E(Z | X = x) = Summation(E(Z | Ai ∩ X = x)P (Ai | X = x) For this part, let Z = the amount of money you have after playing TWO rounds of this “game”. Use the above formula to calculate the conditional expectation of Z given X = $100. (Note: the cards are well-shuffled again between rounds.) (c) Now suppose you get only $16 for drawing a spade. Compute the conditional expectation of Y (not Z) given X = $100
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