An ordinary fair coin is tossed 3 times Outcomes are thus tr

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).

Solution

The probability distribution function of X is as follows:

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

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

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