function fX(x) is a non‐negative function with integral equal to 1 (i.e. the area of the triangle with base of length 4 and height 0.5); hence, it satisfies the conditions in Eqs. (1.12)–(1.14), and it is a valid PDF. We now compute the probability P(2 < X ≤ 3) using the definition in Eq. (1.15):
The probability P(2 < X ≤ 3) represents the area of the region highlighted in Figure 1.2 and can also be computed as the area of a right trapezoid, rotated by 90°.
In many applications, it is useful to describe the distribution of the continuous random variable using the concept of cumulative distribution function (CDF), FX : ℝ → [0, 1]. We assume that a random variable X has a PDF fX(x); then, the CDF FX(x) is defined as the integral of fX(x) in the interval (−∞, x]:
(1.16)
Figure 1.2 Graphical interpretation of the probability P(2 < X ≤ 3) of a continuous random variable X as the definite integral of the PDF in the interval (2, 3].
Using the definition in Eq. (1.15), we conclude that FX(x) is the probability of the random variable X being less than or equal to the outcome x, i.e. FX(x) = P(X ≤ x). As a consequence, we can also define the PDF fX(x) as the derivative of the CDF, i.e.
The CDF takes values in the interval [0, 1] and it is non‐decreasing, i.e. if a < b, then FX(a) ≤ FX(b). When FX(x) is strictly increasing, i.e. if a < b, then FX(a) < FX(b), we can define its inverse function, namely the quantile (or percentile) function. Given a CDF FX(x) = p, the quantile function is
A continuous random variable is completely defined by its PDF (or by its CDF); however, in some special cases, the probability distribution of the continuous random variable can be described by a finite number of parameters. In this case, the PDF is said to be parametric, because it is defined by its parameters. Examples of parameters are the mean and the variance.
The mean is the most common measure used to describe the expected value that a random variable can take. The mean μX of a continuous random variable is defined as:
(1.17)
The mean μX is also called the expected value and it is often indicated as E[X]. However, the mean itself is not sufficient to represent the full probability distribution, because it does not contain any measure of the uncertainty of the outcomes of the random variable. A common measure for the uncertainty is the variance
(1.18)
The variance
Other useful statistical estimators are the median and the mode. The median is the 50th percentile (or P50) defined using the CDF and it is the value of the random variable that separates the lower half of the distribution from the upper half. The mode of a random variable is the value of the random variable that is associated with the maximum value of the PDF. For a unimodal symmetric distribution, the mean, the median, and the mode are equal, but, in the general case, they do not necessarily coincide. For example, in a skewed distribution the larger the skewness, the larger the difference between the mode and the median. For a multimodal distribution, the mode can be a more representative statistical estimator, because the mean might fall in a low probability region. A comparison of these three statistical estimators is given in Figure 1.3, for symmetric unimodal, skewed unimodal, and multimodal distributions.
The parameters of a probability distribution can be estimated from a set of n observations {xi}i=1,…,n. The sample mean
