A coin is tossed 4 times. Let X be the number of heads in the first two tosses and Z be the number of heads in the last three tosses. Describe the joint distribution of (X,Z) by means of a table.
I know how to find the joint distribution of two independent random variables but I'm not sure how you'd go about with this one as the variables are dependent. Do you have to know how they are dependent first? If so, how would I calculate that?
The (X,Z) outcomes are 00, 01, 02, 10, 11, 12, 13, 21, 22, 23. For each of these ten outcomes, determine how many results of the coin tosses generate it. Many correspond to only one result, for example 00 is generated by 0000 only hence its probability is 1/16. Others correspond to several results, for example 11 is generated by 1010, 1001 and 0100 hence its probability is 3/16. These ten probabilities should sum to 1. (To help you check your results, let me mention that six cases have probability 1/16, two have probability 2/16 and two have probability 3/16.)
