## RANDOM VARIABLE AND PROBABILITY

In a clinical study, 1 or more outcomes are observed or measured. Outcomes vary from subject to subject and may or may not be numerical values. For example, the outcome could be "success" or "failure" for a drug treatment or "number of bacteria" in a tissue culture. However, for a statistical analysis, we often want to represent outcomes as numbers.

A random variable is a function that associates a unique numerical value with every possible outcome of a study. There are 2 types of random variables—discrete and continuous.

• A discrete random variable has either a finite or a countable number of distinct values. Examples are as follows

• The number of patients that visited a doctor's office in a day is countable, such as 0, 1, 2, etc

• Of 10 participants in a pilot study, the number of study patients experiencing adverse effects after the treatment must be a finite number of distinct values, 0, 1, 2, …, 10

• A discrete random variable has a probability distribution which provides the possible values of the random variable and their corresponding probabilities. Let P(x) denote the probability that the random variable X equals x. The properties of a probability distribution P(x) are

• 0 ≤ P(x) ≤ 1

• The sum of P(x) for all possible X equals 1.

• A probability distribution can be expressed as a table, graph, or mathematical formula. Examples are as follows

• Suppose a disease can be classified into 5 stages (0–4). The probabilities associated with its classification after a treatment can be expressed as a table (Table 29-1) or a graph (Fig. 29-1)

• The binomial distribution is one of the most common discrete probability distributions in clinical studies. The binomial random variable is defined as the number of "successes" in N independent binary trials such that each trial has 2 possible outcomes, for example, "yes/no" or "success/failure." The probability of success, p, is the same on every trial

• A continuous random variable is one which can take on any value within an interval (or range) of values. It has infinite number of possible values. Continuous random variables are usually measurements such as height, weight, age, or temperature

• A continuous random variable is not defined at specific values. The probability of observing any single value is equal to 0, P(X = x) = 0, since the number of possible values is infinite. Instead, it is defined over a range of values, such as P(X < 5) or P(2 < X < 10) and is equal to the area under a continuous probability distribution curve. The curve, continuous probability distribution f(x), must satisfy the following

• f(x) has no negative values for all x

• The total area under the curve is equal to 1

The most important family of continuous probability distributions is the normal (Gaussian) distribution. It has the following characteristics

• It ...

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