An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of \"heads\

Question

An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails) (t) which we write hth, ttt, etc.? For each outcome, let R be the readom variable counting the number of heads in each outcome. For example, if the outocme id tht, the R(tht)=1. Suppose that the reandom variable X is defined in terms of R as follows: X = 4R - 2R^2 - 1. The values of X are thus:

Calculate the probability distribution function of X, i.e. the function Px (x).

Outcome hth htt thh ttt hht tht tth hhh Value of x -1 1 -1 -1 -1 1 1 -7

Explanation / Answer

The probability distribution function of X is as follows:

PX(x=-1) = P(R=0) + P(R=2) = (0.5)^3 + 3(0.5^2)(0.5) = 0.5

PX(x=1) = P(R=1) = 3(0.5^2)(0.5) = 0.375

PX(x=-7) = P(R=3) = (0.5)^3 = 0.125

Outcome hth htt thh ttt hht tht tth hhh Value of x -1 1 -1 -1 -1 1 1 -7

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