3.1 Random Variables and Discrete Distributions

Overview¶

A random variable is a real-valued function defined on a sample space. Random variables are the main tools used for modeling unknown quantities in statistical analyses. For each random variable $X$ and each set $C$ of real numbers, we could calculate the probability that $X$ takes its value in $C$. The collection of all of these probabilities is the distribution of $X$. There are two major classes of distributions and random variables: discrete (this section) and continuous (3.2 Continuous Distributions). Discrete distributions are those that assign positive probability to at most countably many different values. A discrete distribution can be characterized by its probability mass function (pmf), which specifies the probability that the random variable takes each of the different possible values. A random variable with a discrete distribution will be called a discrete random variable.

3.1.1 Definition of a Random Variable¶

For example, in Example 3.1.1 the number $X$ of heads in the 10 tosses is a random variable. Another random variable in that example is $Y = 10 − X$, the number of tails.

The Distribution of a Random Variable¶

When a probability measure has been specified on the sample space of an experiment, we can determine probabilities associated with the possible values of each random variable $X$. Let $C$ be a subset of the real line such that $\{X \in C\}$ is an event, and let $\Pr(X \in C)$ denote the probability that the value of $X$ will belong to the subset $C$. Then $\Pr(X \in C)$ is equal to the probability that the outcome $s$ of the experiment will be such that $X(s) \in C$. In symbols,

It is a straightforward consequence of the definition of the distribution of $X$ that this distribution is itself a probability measure on the set of real numbers. The set $\{X \in C\}$ will be an event for every set $C$ of real numbers that most readers will be able to imagine.

The general definition of distribution in Definition 3.1.2 is awkward, and it will be useful to find alternative ways to specify the distributions of random variables. In the remainder of this section, we shall introduce a few such alternatives.

Discrete Distributions¶

Random variables that can take every value in an interval are said to have continuous distributions and are discussed in 3.2 Continuous Distributions.

Here are some simple facts about probability mass functions.

A typical pmf is sketched in Figure 3.3, in which each vertical segment represents the value of $f(x)$ corresponding to a possible value $x$. The sum of the heights of the vertical segments in Figure 3.3 must be 1.

Theorem 3.1.2 shows that the pmf of a discrete random variable characterizes its distribution, and it allows us to dispense with the general definition of distribution when we are discussing discrete random variables.

Some random variables have distributions that appear so frequently that the distributions are given names. The random variable $Z$ in Example 3.1.6 is one such example.

The $Z$ in Example 3.1.6 has the Bernoulli distribution with parameter 0.65252. It is easy to see that the name of each Bernoulli distribution is enough to allow us to compute the pmf, which, in turn, allows us to characterize its distribution.

We conclude this section with illustrations of two additional families of discrete distributions that arise often enough to have names.

Uniform Distributions on Integers¶

The $X$ in Example 3.1.8 has the uniform distribution on the integers $0, 1, \ldots, 999$. A uniform distribution on a set of $k$ integers has probability $1/k$ on each integer. If $b > a$, there are $b − a + 1$ integers from $a$ to $b$ including $a$ and $b$. The next result follows immediately from what we have just seen, and it illustrates how the name of the distribution characterizes the distribution.

The uniform distribution on the integers $a, \ldots, b$ represents the outcome of an experiment that is often described by saying that one of the integers $a, \ldots, b$ is chosen at random. In this context, the phrase “at random” means that each of the $b − a + 1$ integers is equally likely to be chosen. In this same sense, it is not possible to choose an integer at random from the set of all positive integers, because it is not possible to assign the same probability to every one of the positive integers and still make the sum of these probabilities equal to 1. In other words, a uniform distribution cannot be assigned to an infinite sequence of possible values, but such a distribution can be assigned to any finite sequence.

Note: Random Variables Can Have the Same Distribution without Being the Same Random Variable. Consider two consecutive daily number draws as in Example 3.1.8. The sample space consists of all 6-tuples $(i_1, \ldots, i_6)$, where the first three coordinates are the numbers drawn on the first day and the last three are the numbers drawn on the second day (all in the order drawn). If $s = (i_1, \ldots, i_6)$, let $X_1(s) = 100i_1 + 10i_2 + i_3$ and let $X_2(s) = 100i_4 + 10i_5 + i_6$. It is easy to see that $X_1$ and $X_2$ are different functions of $s$ and are not the same random variable. Indeed, there is only a small probability that they will take the same value. But they have the same distribution because they assume the same values with the same probabilities. If a businessman has 1000 customers numbered $0, \ldots, 999$, and he selects one at random and records the number $Y$, the distribution of $Y$ will be the same as the distribution of $X_1$ and of $X_2$, but $Y$ is not like $X_1$ or $X_2$ in any other way.

Binomial Distributions¶

Div is a generalization of exm-2-2-5 with $n$ items inspected rather than just six, and rewritten in the notation of random variables. For $x = 0, 1, \ldots, n$, the probability of obtaining each particular ordered sequence of $n$ items containing exactly $x$ defectives and $n − x$ nondefectives is $p^x(1 − p)^{n−x}$, just as it was in exm-2-2-5. Since there are $\binom{n}{x}$ different ordered sequences of this type, it follows that

Therefore, the pmf of $X$ will be as follows:

{#eq-3-1-4}

The reader should be able to verify that the random variable $X$ in Example 3.1.4, the number of heads in a sequence of 10 independent tosses of a fair coin, has the binomial distribution with parameters 10 and $1/2$.

Since the name of each binomial distribution is sufficient to construct its pmf, it follows that the name is enough to identify the distribution. The name of each distribution includes the two parameters. The binomial distributions are very important in probability and statistics and will be discussed further in later chapters of this book.

A short table of values of certain binomial distributions is given at the end of this book. It can be found from this table, for example, that if $X$ has the binomial distribution with parameters $n = 10$ and $p = 0.2$, then $\Pr(X = 5) = 0.0264$ and $\Pr(X \geq 5) = 0.0328$.

As another example, suppose that a clinical trial is being run. Suppose that the probability that a patient recovers from her symptoms during the trial is $p$ and that the probability is $1 − p$ that the patient does not recover. Let $Y$ denote the number of patients who recover out of $n$ independent patients in the trial. Then the distribution of $Y$ is also binomial with parameters $n$ and $p$. Indeed, consider a general experiment that consists of observing $n$ independent repititions (trials) with only two possible results for each trial. For convenience, call the two possible results “success” and “failure.” Then the distribution of the number of trials that result in success will be binomial with parameters $n$ and $p$, where $p$ is the probability of success on each trial.

Note: Names of Distributions. In this section, we gave names to several families of distributions. The name of each distribution includes any numerical parameters that are part of the definition. For example, the random variable $X$ in Example 3.1.4 has the binomial distribution with parameters 10 and $1/2$. It is a correct statement to say that $X$ has a binomial distribution or that $X$ has a discrete distribution, but such statements are only partial descriptions of the distribution of $X$. Such statements are not sufficient to name the distribution of $X$, and hence they are not sufficient as answers to the question “What is the distribution of $X$?” The same considerations apply to all of the named distributions that we introduce elsewhere in the book. When attempting to specify the distribution of a random variable by giving its name, one must give the full name, including the values of any parameters. Only the full name is sufficient for determining the distribution.

Summary¶

A random variable is a real-valued function defined on a sample space. The distribution of a random variable $X$ is the collection of all probabilities $\Pr(X \in C)$ for all subsets $C$ of the real numbers such that $\{X \in C\}$ is an event. A random variable $X$ is discrete if there are at most countably many possible values for $X$. In this case, the distribution of $X$ can be characterized by the probability mass function pmf of $X$, namely, $f(x) = \Pr(X = x)$ for $x$ in the set of possible values. Some distributions are so famous that they have names. One collection of such named distributions is the collection of uniform distributions on finite sets of integers. A more famous collection is the collection of binomial distributions whose parameters are $n$ and $p$, where $n$ is a positive integer and $0 < p < 1$, having pmf eq-3-1-4. The binomial distribution with parameters $n = 1$ and $p$ is also called the Bernoulli distribution with parameter $p$. The names of these distributions also characterize the distributions.

Exercises¶