Final answer:
The question relates to probability theory, focusing on the discrete probability density function (PDF) and cumulative distribution function (CDF) of a random variable. The PDF for a discrete, countable variable conforms to probabilities summing to one, while the CDF helps find the probability of a variable being less than or more than a certain value.
Step-by-step explanation:
Understanding Probability Distribution Functions
The question delves into the domain of probability theory, specifically focusing on the concepts of the probability density function (PDF) for a discrete random variable, and the cumulative distribution function (CDF). When dealing with a discrete random variable like X, which takes on countable values such as 0, 1, 2, 3, 4, and 5, the PDF is characterized by two main properties:
By applying these properties, a PDF can be constructed for a variable like X representing the number of days the men's soccer team plays per week.
The CDF, P(X ≤ x), represents the probability of the random variable being less than or equal to a certain value x. This function is particularly useful in continuous probability distributions to determine the likelihood that a variable falls within a certain range. To find the probability that a variable exceeds a certain value, one can use the CDF by calculating 1 − P(X < x).
To summarize, defining X, listing possible values, constructing a PDF, and understanding the CDF are key steps in tackling probability distribution problems.