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The Cumulative Distribution Function (CDF) is an essential concept in probability that tells us the probability that a random variable is less than or equal to a certain value. In simpler terms, it helps us understand the accumulation of probabilities up to a specific point within a probability distribution.
For the random variable in our exercise, which is the maximum roll of two dice, the CDF is a way to show, for each possible maximum outcome, the summed probabilities of observing this maximum or any lower value.
The CDF of the maximum roll on two dice, denoted as \( F(m) \), is calculated by adding up the probabilities from the Probability Mass Function (PMF) up to the value \( m \).
For example:

• \( F(1) = \frac{1}{36} \)
• \( F(2) = \frac{1}{36} + \frac{1}{12} = \frac{4}{36} \)
• And so on, until \( F(6) = 1 \), indicating that you will always roll a number between 1 and 6

The CDF is particularly useful for understanding probabilities in a cumulative manner and is often depicted as a step function, showcasing the sum of the PMF values as you move up the possible outcomes of a discrete random variable.

The Probability Mass Function (PMF) is a function that provides the probabilities of all possible discrete outcomes of a random variable. It essentially maps each outcome to its likelihood, which is essential in understanding the probability distribution of discrete random variables like rolling dice.
In our exercise, the PMF describes the probability of the maximum number shown when two six-sided dice are rolled. This is given by \( p(m) \), where \( m \) is the possible maximum value:

• \( p(1) = \frac{1}{36} \)
• \( p(2) = \frac{1}{12} \)
• \( p(3) = \frac{5}{36} \)
• \( p(4) = \frac{7}{36} \)
• \( p(5) = \frac{1}{4} \)
• \( p(6) = \frac{11}{36} \)

These probabilities add up to 1, as they encompass all possible maximum outcomes when two dice are rolled. The PMF is vital for clearly specifying the likelihood of each possible outcome, allowing us to further compute cumulative probabilities and draw probability-related insights.

A Discrete Random Variable is a type of variable that can take on a finite or countable number of values, each associated with a probability. They are the cornerstone of probability theory when dealing with real-world applications like games involving dice or cards.
In our exercise, the discrete random variable is \( M \), representing the maximum outcome when two dice are rolled.
Since each die can show a number between 1 to 6, the possible values for \( M \) range from 1 to 6 as well.

• \( M = 1 \) indicates both dice show a 1
• \( M = 2 \) occurs if at least one die shows a 2, and neither die shows more than 2
• ...continuing up to \( M = 6 \), which is the highest possible maximum outcome

This highlights how discrete random variables quantify outcomes in scenarios where results are distinct and separate, forming the basis for more complex probability measures like the PMF and CDF.