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# Probability distribution function

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Please see the attached file for full problem description with all proper "X" symbols.

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Can you show me how to complete this kind of problem? Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" ( ) and "boys" ( ), which we write , , etc. For each outcome, let be the random variable counting the number of boys in each outcome. For example, if the outcome is , then . Suppose that the random variable is defined in terms of as follows: . The values of 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 , i.e. the function . First, fill in the first row with the values of , in increasing order. Then fill in the appropriate probabilities in the second row. Enter each probability as an exact fraction.