Random Variable

A variable whose observed values may be considered as outcomes of a stochastic or random experiment is called a random variable. The values of such a variable in a particular sample cannot be anticipated with certainty before the sample is gathered. Random variables are commonly classified as qualitative or categorical, discrete, or continuous.

A random variable is defined as a qualitative or categorical variable if its set of possible values do not represent numerical information. For example, gender is a categorical variable. Suppose that among 100 patients, there are 65 females and 35 males, and let X be the sex of a randomly chosen patient among these 100 patients. Then X is a qualitative random variable, and the values of X are “Female” and “Male” with 65% and 35% chance to be chosen, respectively. Even if numeric values, such as 0 and 1, are used to code gender in a data set, it remains a categorical variable because the values represent membership in a category rather than a measured quantity.

A random variable is discrete if its set of possible values is countable. If only two values are possible, such as alive versus dead, it may also be called a binomial random variable. For example, a new technique, balloon angioplasty, is being widely used to open clogged heart valves and vessels. The balloon is inserted via a catheter and is inflated, opening the vessel; thus, no surgery is required. Suppose that among untreated people with heart-valve disease, about 50% die within 2 years, and experience with balloon angioplasty suggests that approximately 70% treated with this technique live for more than 2 years. We can define X as the number of patients who will live more than 2 years, among the next five patients treated with balloon angioplasty at a hospital. Then, X constitutes a discrete random variable, which can take on the values 0, 1, 2, 3, 4, or 5.

To make an inference about the population from our sample data, we need to know the probability associated with each value of the variable that is called its probability distribution. Probability calculations are relatively simple for discrete variables and are often displayed in tabular form, as presented below. The probability distribution for a discrete random variable X displays the probability P(x) associated with each value of x: This display can be presented as a table, a graph, or a formula. To illustrate, consider the above example; all possible values of X are 0, 1, 2, 3, 4, and 5. The probability distribution is a binomial distribution with n = 5 and p = 0.7 that can be given by the formula:

It may also be displayed as shown in Table 1.

Figure 1 Bar Chart Displaying Probability Distribution for Discrete Random Variable X

Or, it may be presented graphically as a bar chart, as shown in Figure 1.

The properties of discrete random variables are as follows:

• The probability associated with every value of x lies between 0 and 1.
• The sum of the probabilities for all values of x is equal to 1.
• The probabilities are additive, that is, P(X ≥ 4) is the same P(X = 4)+P(X = 5).

A random variable is defined as continuous if its set of possible values is an entire interval on the number line, that is, if it can take any value within a range rather than only a discrete set of values such [Page 893]as was specified in the previous example. Of course, any measuring device has a limited accuracy and therefore a continuous scale may in practice be something of an abstraction. Some examples of continuous random variables are the height of an adult male, the weight of a newborn baby, a patient's body temperature, and the survival time of a patient following a heart attack.

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