Consider a random experiment of tossing a biased coin three times. We assume tha

Question

Consider a random experiment of tossing a biased coin three times. We assume that the tosses are indepen- dent of each other and the probability of a head is 0.2. Let X be the random variable whose value represent:s the number of heads obtained (a) (6 pts) List the possible outcomes of three coin tosses. What are the probabilities PrX 0], Pr[X1], Pr(X = 2]. Prix = 3)? variables). Show your work. Define X in terms of X's and compute E[X using this expression. Show your work. (b) (12 pts) Compute E[Xusing the definition of expected value (i.e., without the use of indicator random (c) (12 pts) Let Xi be an indicator random variable, whose value is 1 if the i-th toss resulted in heads.

Explanation / Answer

a. The possible outcomes of the three tosses are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT

P(X=0) = (0.8)3 = 0.512

P(X=1) = 3*0.2*0.82 = 0.384

P(X=2) = 3*0.22*0.8 = 0.096

P(X=3) = 0.23 = 0.008

b. E(X) = 0*P(X=0) + 1*P(X=1) + 2*P(X=2) + 3*P(X=3) = 0 + 0.384*1 + 0.096*2 + 0.008*3 = 0.6

c. Xi denotes 1 if the ith toss is heads. So, X = X1 + X2 + X3

So, E(X) = E(X1 + X2 + X3)

E(X) = E(X1)+ E(X2) + E(X3)

Now, E(X1) = E(X2) = E(X3) = 0*0.8 + 1*0.2 = 0.2

So, E(X) = 3*E(X1) = 3*0.2 = 0.6

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