Final answer:
The graphical space of outcomes of rolling two four-sided dice comprises x ranging from 1 to 4 and y from 2 to 8. Their joint pmf is evenly distributed, the marginal pmf of x is 1/4 for each value, and the marginal pmf of y varies. X and y are dependent as y's value is influenced by x.
Step-by-step explanation:
When we roll a pair of four-sided dice, with one red and one black, we can describe the space of possible outcomes (x, y) where x is the outcome of the red die and y is the sum of the outcomes on both dice. Graphing this on graph paper would result in a space where x can range from 1 to 4 and y can range from 2 to 8, as the smallest sum occurs when both dice show 1 and the largest sum occurs when both dice show 4.
The joint probability mass function (pmf) would be defined on this space, assigning equal probability to each outcome pair since we are dealing with fair dice. However, the marginal pmf of x would be the probability distribution of the outcomes on the red die alone, which, being a fair four-sided die, would be 1/4 for each of its possible values (1 to 4).
The marginal pmf of y would be the probability distribution for the sum of the two dice. To find these probabilities, we would count the number of ways each sum can occur and divide by the total number of outcomes (16, as there are 4 possibilities for the red die and 4 for the black die).
Lastly, variables x and y are dependent because the value of y is partially determined by the value of x (since y is the sum and includes the outcome of x). If x is high, the minimum value of y is constrained to be higher as well.