Fundamentals of Probability and Statistics for Engineers

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Part A Probability and R andom Variables ...

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2 BASIC PROBABILITY CONCEPTS The mathematical theory of probability gives us the basic tools for constructing and analyzing math ...

2.1 Elements of Set Theory Our interest in the study of a random phenomenon is in the statements we can make concerning the even ...

Example 2.1.Let and Then since every element of is also an element of. This relationship can also be presented graphically by us ...

The intersection or product of A and B, written as A B, or simply AB, is the set of all elements that are common to A and B. In ...

stands for the set of all elements belonging to one or more of the sets Aj, j 1,2,...,n.Theintersection is the set of all elemen ...

The second relation in Equations (2.10) gives the union of two sets in terms of the union of two disjoint sets. As we will see, ...

A,B,C,.... Some of these corresponding sets and probability meanings are given in Table 2.1. As Table 2.1 shows, the empty set i ...

Axiom 1: P(A) 0 (nonnegative). Axiom 2: P(S) 1 (normed). Axiom 3: for a countable co llection of mutually exclusive events A 1 , ...

Hence, using Axiom 3, Furthermore, we note Hence, again using Axiom 3, or Substitution of this equation into Equation (2.13) yie ...

and, in the case of n events, where Aj, j 1,2,...,n, are arbitrary events. Example 2.5.Let us go back to Example 2.4 and assume ...

probability of ‘heads’ in flipping a coin is 1/2. The relative frequency approach to probability assignment is objective and con ...

we expect that the ratio nAB/nA also tends to P(B) as nA becomes large. The independence assumption then leads to the observatio ...

where B 1 ,B 2 ,andB 3 are mutually exclusive, each occurring with probability^14. It is easy to calculate the following: We see ...

An expression for q may also be obtained by noting that the system fails if any one or more of the five components fail, or wher ...

D ef inition 2. 3. The conditional probability of A given that B has occurred is given by Definition 2.3 is meaningless if P(B) ...

and the third axiom holds. The definition of conditional probability given by Equation (2.24) can be used not only to compute co ...

Equation (2.25) is seen to be useful for finding joint probabilities. Its exten- sion to more than two events has the form where ...