Page 271 - Schaum's Outlines - Probability, Random Variables And Random Processes

Chapter 8

Decision Theory

8.1 INTRODUCTION
There are many situations in which we have to make decisions based on observations or data
that are random variables. The theory behind the solutions for these situations is known as decision
theory or hypothesis testing. In communication or radar technology, decision theory or hypothesis
testing is known as (signal) detection theory. In this chapter we present a brief review of the binary
decision theory and various decision tests.

8.2 HYPOTHESIS TESTING
A. Definitions:
A statistical hypothesis is an assumption about the probability law of r.v.'s. Suppose we observe a
random sample (XI, . . . , X,) of a r.v. X whose pdf f (x; 0) = f (x,, . . . , x,; 8) depends on a parameter 8.
We wish to test the assumption 8 = 8, against the assumption 8 = 8,. The assumption 8 = 8, is
denoted by H, and is called the null hypothesis. The assumption 8 = 8, is denoted by H, and is called
the alternative hypothesis.
H,: 8 = 8, (Null hypothesis)
H, : 8 = 8, (Alternative hypothesis)
A hypothesis is called simple if all parameters are specified exactly. Otherwise it is called compos-
ite. Thus, suppose H,: 0 = 8, and H, : 8 # 0,; then H, is simple and H, is composite.

B. Hypothesis Testing and Types of Errors:
Hypothesis testing is a decision process establishing the validity of a hypothesis. We can think of
the decision process as dividing the observation space Rn (Euclidean n-space) into two regions R, and
R,. Let x = (x,, . . . , x,) be the observed vector. Then if x E R,, we will decide on H,; if x E R,, we
decide on H,. The region R, is known as the acceptance region and the region R, as the rejection (or
critical) region (since the null hypothesis is rejected). Thus, with the observation vector (or data), one
of the following four actions can happen:
1. H, true;accept H,
2. H, true; reject H, (or accept H,)
3. H, true; accept H,
4. H, true; reject H, (or accept H,)
The first and third actions correspond to correct decisions, and the second and fourth actions corre-
spond to errors. The errors are classified as
1. Type I error: Reject H, (or accept H,) when H, is true.
2. Type I1 error: Reject H, (or accept H,) when H, is true.
Let PI and PI, denote, respectively, the probabilities of Type I and Type I1 errors: