Final answer:
The random variable X represents the number of freshmen asked before receiving a 'yes' response in a geometric distribution scenario. X can take any positive integer value, and the geometric distribution formula is used to construct the probability distribution function. Probability of success is given as 0.713 for this scenario.
Step-by-step explanation:
Geometric Distribution in Probability
The random variable X in this context represents the number of first-time, full-time freshmen that must be asked until finding one who agrees with a recent law that was passed, termed as a 'success'.
Given that 71.3 percent agreed with the law, the geometric distribution is suitable to model this random variable since it describes the probability of having a certain number of failures before the first success in a sequence of Bernoulli trials, where each trial is independent and has the same probability of success.
To define X, one would say it is the count of freshmen asked before a 'yes' response is received. X can take on any positive integer value, starting at 1, since you need to ask at least one person. To construct the probability distribution function (PDF) for X, you would use the geometric distribution formula P(X = x) = (1 - p)^(x - 1) * p, where p is the probability of success on each trial. For this example, p = 0.713, and you would calculate the probabilities for X = 1, 2, 3, ..., 6 accordingly.
For a more complete answer, the student would need to provide the full joint probability density function for random variables mentioned, as well as any given values or parameters, in order to find specific probabilities, means, standard deviations, and covariances.