Question

3. (25 pts) A Truly FAIR COIN: Because actual coins are not truly balanced, P - the ACTUAL probability of HEAD for our old, battered coin - may differ substantially from 1/2. The famous Mathematician John Von-Neumann came up with the following proposal for using our possibly unfair coin to simulate a truly fair coin that always has PROB(HEAD)=PROB(TAIL) = 1/2, as follows: • (i) toss the UNFAIR coin twice. This is the experiment E. • (ii) IF you got HT, SUCCESS, and you report “HEAD” as the outcome and STOP • (iii) IF you got TH, also SUCCESS, but now you report “TAIL” as the outcome and STOP • (iv) IF you got either HH or TT, E was a FAILURE, so NO RESULT can be reported. Instead, GO back to (i) and perform E again (repeating as many times as needed until you get SUCCESS). Let’s suppose your old coin had PROB(Head)= 2/5 and PROB(Tail)= 3/5. (a) FIRST: does the von-Neumann algorithm really report HEAD and TAIL, each with probability 1/2? Explain why you said YES or NO. (b) Let N1 be the random variable that counts the number of times you perform E in the above algorithm in order to generate ONE SUCCESS. What is E(N1), the expectation? What is V (N1), the variance? Again, explain your answers

3. (25 pts) A Truly FAIR COIN: Because actual coins are not truly balanced, P - the ACTUAL probability of HEAD for our old, battered coin may differ substantially from 1/2. The famous Mathematician John Von-Neumann came up with the following proposal for using our possibly unfair coin to simulate a truly fair coin that always has PROB( HEAD-PROB(TAIL) = 1/2, as follows: . (i) toss the UNFAIR coin twice. This is the experiment £. . (ii) IF you got HT, SUCCESS, and you report "HEAD" as the outcome and STOP (iii) IF you got TH, also SUCCESS, but now you report "TAIL" as the outcome and STOP (iv) IF you got either HH or TT, was a FAILURE, so NO RESULT can be reported. Instead GO back to (i) and perform E again (repeating as many times as needed until you get SUCCESS) 1V Let's suppose your old coin had PROB(Head)- 2/5 and PROB (Tail) 3/5. (a) FIRST: does the von-Neumann algorithm really report HEAD and TAIL, each with probability (b) Let Ni be the random variable that counts the number of times you perform & in the above 1/2? Explain why you said YES or NO algorithm in order to generate ONE SUCCESS. What is E(Ni), the expectation? What is V (Ni), the variance? Again, explain your answers

Answer 1