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1. The probability that an unfair coin lands on heads is 0.6. Coin is tossed 3 times.
a. List the sample space.
b. Construct a random variable X which counts number of heads for the three tosses.
c. Find P(X<1), E(X) and E(1/(X+1))
Is this the sample space? {HHH, THH, HTH, HHT, THT, TTH, HTT, TTT}.
a.
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
P (HHH) = .216
P (HHT) = P (HTH) = P (THH) = .144
P (HTT) = P (THT) = P (TTH) = .096
P (TTT) = .064
.216 + (3).144 + (3).096 + .064 = 1
b.
P (X = 3) = .216
P (X = 2) = .432
P (X = 1) = .288
P (X = 0) = .064
.216 + .432 + .288 + .064 = 1
c.
P (X<1) = P (X=0) = .064
E (X) = (3*(.216)) + (2*(.432)) + (1*(.288)) + (0*(.064)) = 1.8
Good work! From the distribution, we can see a shifting of the middle towards the right (and skewing to the left). With a fair coin, the mean would be 1.5 rather than 1.8.
Visual representation of the problem as a tree
For the second part of (c), E(1/(X+1))
E(1/(X+1))=0.064+(0.288/2)+(0.432/3)+(0.216/4)=0.406
sorry for the rough sketch quality
well looking at these solution from other persons are really helping to figure out where i went wrong. NICE WORK mendozak
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