Calculate the probability distribution function of x

Question

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Q: Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" ( g) and "boys" (b ), which we write gbg , bbb , etc. For each outcome, let R be the random variable counting the number of boys in each outcome. For example, if the outcome is gbb , then Rgbb=2 . Suppose that X the random variable is defined in terms of R as follows: X=R²-3R-4

The values of X are thus:

Outcome bbg bgg gbb bbb gbg ggb bgb ggg

Value of X -6 -6 -6 -4 -6 -6 -6 -4

Calculate the probability distribution function of X, i.e. the function P_X(X) . First, fill in the first row with the values of X, in increasing order. Then fill in the appropriate probabilities in the second row. Enter each probability as an exact fraction.