Question 4: You repeatedly flip a fair coin and stop as soon as you get tails...

 

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Question 4: You repeatedly flip a fair coin and stop as soon as you get tails followed by heads. (All coin flips are mutually independent.) Consider the random variable X = the total number of coin flips. For example, if the sequence of coin flips is HHHTTTTH, then X = 8. • Determine the expected value E(X) of X. Hint: Use the linearity of expectation. You may use any result that was proven in class. Question 5: Let X and Y be two independent random variables on the same sample space. We make the following assumptions: 1. X and Y have the same probability distribution, i.e., for all k, Pr(X= k) = Pr(Y = k). Note that this implies that E(X)=E(Y). 2. For any element in the sample space, the values of both X and Y are non-negative. Consider the two random variables and Y' = y². • Let a and b be two non-negative real numbers. Prove that min (a², b²) ≤ ab. Prove that X' = X² E (min (X', Y')) ≤ (E(X))², E (min (X², Y²)) ≤ (E(X))², Hint: Since X and Y have the same probability distribution, E(X) = E(Y). Since X and Y are independent, E(XY) = E(X). E(Y). i.e., Question 4: You repeatedly flip a fair coin and stop as soon as you get tails followed by heads. (All coin flips are mutually independent.) Consider the random variable X = the total number of coin flips. For example, if the sequence of coin flips is HHHTTTTH, then X = 8. • Determine the expected value E(X) of X. Hint: Use the linearity of expectation. You may use any result that was proven in class. Question 5: Let X and Y be two independent random variables on the same sample space. We make the following assumptions: 1. X and Y have the same probability distribution, i.e., for all k, Pr(X= k) = Pr(Y = k). Note that this implies that E(X)=E(Y). 2. For any element in the sample space, the values of both X and Y are non-negative. Consider the two random variables and Y' = y². • Let a and b be two non-negative real numbers. Prove that min (a², b²) ≤ ab. Prove that X' = X² E (min (X', Y')) ≤ (E(X))², E (min (X², Y²)) ≤ (E(X))², Hint: Since X and Y have the same probability distribution, E(X) = E(Y). Since X and Y are independent, E(XY) = E(X). E(Y). i.e.,

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