Basic Probability Theory for Biomedical Engineers - JohnD. Enderle

92 BASIC PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS

Definition 2.3.5. A RV x defined on (S, F, P ) is continuous if the CDF Fx is absolutely continuous. To avoid technicalities, we simply note that if Fx is absolutely continuous then Fx is continuous everywhere and Fx is differentiable except perhaps at isolated points. Consequently, there exists a function fx satisfying

The function fx is called the probability density function (PDF) for the continuous RV x. The set of

α−h

= fx (α).

The above is a special case of Leibnitz’ rule.

The PDF fx is nonnegative since the CDF Fx is monotone nondecreasing. Since Fx (+∞) = 1, we have

∞

1 = fx (α ) d α .

−∞

where the limit is through positive values of h. For a continuous RV, the rightand left-hand derivatives are equal almost everywhere; however, a treatment of discrete and mixed RVs using a PDF containing Dirac delta functions can be developed. Such a treatment is offered in the following section, with the Dirac delta function

where the limit requires a special interpretation.

Example 2.3.5. The PDF for the RV x is

βα2, −1 < α < 2 fx (α) = 0, otherwise.

Find β so that fx is a PDF, and find the CDF Fx .

Solution. We require

so that β = 1/3. We note that fx (α) ≥ 0, as required. We obtain the CDF using

α

Fx (α) = fx (α ) d α .

−∞

Since fx (α ) = 0 for α < −1 we obtain Fx (α) = 0 for α < −1. For −1 ≤ α < 2 we obtain

94 BASIC PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS

fx (α )

4

3

1

3

could be defined to be any real numbers without affecting the result. In fact, the PDF could be redefined at any discrete set of points without affecting the result.

The PDF and CDF for this example are illustrated in Fig. 2.5. It is extremely important

2.3.3Mixed Random Variables

Mixed random variables are neither discrete nor continuous. The resulting CDF is piecewise continuous. Probabilities for mixed random variables can be found in several ways. Splitting the CDF into the sum of a jump CDF and a continuous CDF and using the corresponding PMF and PDF is one approach, and is treated in this section.

RANDOM VARIABLES 95

Definition 2.3.6. A RV x defined on (S, F, P ) is a mixed RV if it is neither discrete nor continuous.

Theorem 2.3.4 (Lebesgue Decomposition Theorem). A CDF Fmay be expressed as

where 0 ≤ γ ≤ 1, FC is a continuous CDF, and FD is a discrete CDF.

Proof. Note that if F is continuous then γ = 1 and FC = F. Similarly, if F is discrete then γ = 0 and FD = F.

Assume F is neither continuous nor discrete. Define

q (α) = F(α) − F(α−).

Then q (α) ≥ 0, and q (α) = 0 only at isolated points, say α D. Let

(1 − γ )FD(α) = q (α )

α ≤α

and

1 − γ = q (α ).

α D

Then FD is a monotone nondecreasing, right-continuous jump function, with FD(−∞) = 0 and FD(+∞) = 1; i.e., FD is a discrete CDF. Now, let

FC (α) = F(α) − (1 − γ )FD(α) .

γ

Then FC (−∞) = 0, FC (+∞) = 1, and FC is right-continuous since both F and FD are right-continuous. Also,

Although the above decomposition theorem is useful, there is no guarantee that FC is absolutely continuous and hence that FC is the CDF of a continuous RV. It can be shown that FC can always be further decomposed into the sum of an absolutely continuous part and a singular part [4]. We will assume throughout that the singular part is zero, and hence that FC is the CDF for a continuous RV. All CDFs arising in practical applications satisfy this assumption. For our purposes then, if γ = 1 the CDF describes a continuous RV and if γ = 0 the CDF describes a discrete RV.

RANDOM VARIABLES 97

Fx (α )

1

0.5

Drill Problem 2.3.1. The discrete random variable x has cumulative distribution function Fx shown in Fig. 2.7. Find (a) px (−1), (b) px (0), (c )P (0 ≤ x ≤ 3), (d )P (0 < x ≤ 2).

Answers: 7/8, 1/8, 0, 1/2.

98 BASIC PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS

Drill Problem 2.3.2. A committee of three members is to be formed from four engineers and three physicists. Let x be a RV which assigns to every sample point in S a value equal to the number of engineers on the committee. Determine: (a) px (0), (b) px (1), (c ) px (2), and (d) px (3).

Answers: 1/35, 4/35, 12/35, 18/35.

Drill Problem 2.3.3. The waveform w(t) is uniformly sampled every 0.2 s from t = 0 s to t = 4 s, where

The sampled values are rounded off to the nearest integer and collected in the set S. The RV x(ζ ) = ζ for all ζ S. Determine: (a) px (0), (b) px (1), (c ) px (2), (d ) px (3), (e ) px (5).

Answers: 1/3, 5/21, 4/21, 1/7, 0.

Drill Problem 2.3.4. Suppose the RV x has the PMF

Find: (a) Fx (−1), (b)Fx (0), (c )Fx (1), (d )Fx (3).

Answers: 3/8, 1/8, 3/4, 0.

Drill Problem 2.3.5. The PDF for the RV x is

where β is a constant. Determine: (a) β, (b)P (x ≤ −1), (c )Fx (0), (d )P (0 ≤ x ≤ 2).

Answers: 0, 1/9, 2/9, 8/9.

where β is a constant. Determine: (a) β, (b)P (x ≥ 1/2), (c )Fx (1/4), (d )P (x = 1/4).

Answers: 0.406, 3/8, 0, 0.381.

100 BASIC PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS

where F is a CDF, a and b are real numbers, B is a Borel set, and g : R →R . Such integrals are known as Riemann-Stieltjes integrals. In the following we assume that F is the CDF for the RV x and that a < b. In the special case that B = (a, b] and g (α) = 1 for all α, then all the above integrals are the same. We establish below that

a

The Riemann-Stieltjes integral provides a unified framework for treating continuous, discrete, and mixed RVs—all with one kind of integration. An important alternative is to use a standard Riemann integral for continuous RVs, a summation for discrete RVs, and a Riemann integral with an integrand containing Dirac delta functions for mixed RVs.

We begin with a brief review of the standard Riemann integral. Let

provided the limit exists and is independent of the choice of {ξi }. Note that n → ∞ as n → 0. The summation above is called a Riemann sum. We remind the reader that this is the “usual” integral of calculus and has the interpretation as the area under the curve h between a and b.

With the same notation as above, the Riemann-Stieltjes integral is defined by

provided the limit exists and is independent of the choice of {ξi }.