Final answer:
A random variable X can be used to describe the weather status and temperature for Arlington in April with six equally probable outcomes. The Probability Mass Function (PMF) each would be 1/6, and the Cumulative Distribution Function (CDF) would step up by 1/6 for each additional outcome.
Step-by-step explanation:
To define a random variable for a day's weather status and temperature in April in Arlington, you could let X represent the combined outcome of weather status and temperature range. For weather status, there are three possibilities: sunny, rainy, or cloudy. For temperature range, there are two possibilities: 70-85 or 86-100 degrees Fahrenheit. Therefore, X can take on one of six possible values: (Sunny, 70-85), (Sunny, 86-100), (Rainy, 70-85), (Rainy, 86-100), (Cloudy, 70-85), (Cloudy, 86-100).
The Probability Mass Function (PMF) of X would assign a probability of 1/6 to each of these six outcomes because each outcome of weather status (three possible outcomes) is equally likely, and each outcome of temperature range (two possible outcomes) is equally likely, and the two events are independent. Hence, the product of their probabilities is 1/3 * 1/2 = 1/6.
The Cumulative Distribution Function (CDF) of X would look like a step function, where the probability increases by 1/6 for each of the possible outcomes of X. It would start at 0, and at the first outcome of X (Sunny, 70-85), the CDF would be 1/6. Then at the next outcome (Sunny, 86-100), it would increase to 2/6, and so on, until at the last outcome (Cloudy, 86-100), the CDF would be 1.