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A discrete random variable is one that can take on a countable number of distinct values. This means that the values can be listed out. For example, the number of phone calls received in a day is a discrete random variable because you can count the calls as 0, 1, 2, etc.
A discrete random variable contrasts with a continuous random variable, which can take on infinitely many values within a range. For instance, temperature readings are continuous because they can be any value within a range.
Since the random variable in our exercise represents the number of phone calls, it takes on discrete, countable values. Non-integer values like 2.3, therefore, are not possible for this variable. The value must be a whole number, aligning it with the definition of a discrete random variable.

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. The Poisson distribution is one type of probability distribution specifically used for discrete random variables. It gives the probability of a number of events happening in a fixed interval of time or space, given a known average rate.
In our exercise, the random variable follows a Poisson distribution with a mean (average rate) of 7.2 calls per day. This distribution helps us understand and predict the likelihood of receiving any specific number of phone calls within a day.
The key characteristic of a Poisson distribution is that it only deals with non-negative integer values, which means it counts occurrences like 0, 1, 2, 3, etc. It cannot provide probabilities for non-integer values like 2.3.

Since our random variable represents the number of phone calls received in a day, the values it can take are limited to whole numbers. These values are integer values, which are numbers without any fractional parts.
The set of integer values includes numbers like 0, 1, 2, 3, and so forth. These values represent the countable aspects of the random variable in question. Non-integer values such as 2.3 do not fit within this framework because they cannot represent discrete counts of events.
Understanding that a Poisson-distributed variable only takes integer values is crucial for correctly interpreting and working with such distributions. This knowledge helps in performing accurate statistical analysis and solving related problems.