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Two coins have been rigged so that when they are tossed, the second coins always lands tails up when the first coin lands heads up and vice versa....
Two coins have been rigged so that when they are tossed, the second coins always lands tails up when the first coin lands heads up and vice versa.
The probability of each coin landing heads up is 0.5
Define random variables X and Y as follows.
X = 1 if first coin lands heads up
= 0 if first coin lands tails up
Y = 1 if second coin lands heads up
= 0 if second coin lands tails up
a) Calculate E(X) and Var(X)
b) Calculate COV(X, Y) and ρ(X, Y)
c) Write down the distribution - ie, values and probabilities - of X + Y
d) Using your answer to c), calculate E(X + Y) and Var(X + Y)
e) Based on the answers to the previous parts, show that:
E(X + Y) = E(X) + E(Y)
Var(X + Y) = Var(X) + Var(Y) + 2 COV(X, Y)
(Note: You are NOT allowed to use the formulas in part e) to answer part d). If you do so, you will get a zero for both part d) and part e))
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