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# Discrete in Probability

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One area of probability and mathematical statistics deals with discrete random variables. The word discrete is unfamiliar to some, but it is important to know what sorts of things are described by this term. Discrete mathematics deals with those things that can be separated from one another and counted. Discrete objects are of a different nature than those topics in mathematics that are continuous. These concepts require some more careful explanation.

### Density and Continuity

For much of mathematics we work with the set of real numbers. This set has many important features that make it useful and applicable to any number of problems. One of these features is that the set of real numbers is dense. This means that between any two real numbers, there is another real number.

The feature of density may seem simple, but it goes a long way. If our first two real numbers are x and y, there exists a real number z between these numbers. In turn, there is another real number, call it w between x and z. Keep continuing in this fashion. Since we can always find a real number between any two real numbers, there are infinitely many real numbers between any two real numbers. The set of all real numbers can be thought of as a number line with no gaps or holes in it.

### Contrast with Discrete

Not everything in mathematics is continuous. In contrast with continuous mathematical topics are those that are discrete. Discrete objects in mathematics are those things that can be separated from each other. The real number line is not a discrete set, but other sets commonly used in mathematics are discrete. For example, the following are discrete:

• Whole numbers {0, 1, 2, 3, . . .}
• Integers {. . .-3, -2, -1, 0, 1, 2, 3, . . . }
To recall a key term above, these sets are not dense. For each of these sets there we can find two numbers that have no other number from the set that is between them. Consider the numbers 0 and 1, which are elements of both sets. There are no whole numbers or integers between these two numbers.

### Applications of Discrete Mathematics

Although there are some areas of mathematics, such as calculus, that require concepts of continuity, discrete mathematics is a very important topic. Whenever objects are counted, discrete mathematics is being used. Other areas of mathematics that involve using discrete ideas or discrete sets include:

• The area of mathematics known as combinatorics, which involves counting problems involving combinations and permutations
• The study of relationships and functions between sets with finite numbers of elements
• The analysis of processes with a finite number of steps. This includes the study of algorithms and programs that are used in computer science
• Graph theory is an area of mathematics that explores how finite sets of objects are connected to one another.
• Logic and the study of the structure of mathematical proofs
• Probability distributions involving experiments with a finite set of outcomes

### Discrete Probability Distributions

Discrete ideas are used in probability and statistics when experiments have a finite set of outcomes. Since there are only a finite number of probabilities obtained, the graph of a discrete probability distribution is not a smooth curve. The function that defines this distribution looks like a series of stair steps, and is not a continuous function. Instead of some sort of area underneath a smooth continuous curve, the distribution for a discrete random variable is a histogram. One very important example of the discrete in probability is the binomial distribution. This distribution is used when there is a probability experiment with a fixed number of finite trials.

Courtney Taylor
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