This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Page 1 Probability of Independent Events • Definition: – Events A and B are i ndependent if the occurrence of event B does not alter the occurrence of event A P(A I B) = P(A) P(A B) = P(A)P(B) Proof: • Brief review: Consider a system with two failure modes. Let A = { a: system fails mode 1}; B = { b: system fails mode 2} A’ = { a: system does not fail mode 1}; B’ = { b: system does not fail mode 2} U P(A I B) ≝ ( ? ? ) ( ? ) = P(A) Page 2 Probability of Independent Events (Con’t) • Brief review (Con’t): – The probability the system fails is The probability that the system does not fail is – If the failure modes are independent P(A U B) = P(A) + P(B)  P(A) P(B) A’ and B’ are also independent = 1  P(A U B) – If failure modes are mutually exclusive P(A U B) = P(A) + P(B) P(A U B) = P(A) + P(B)  P(A B) P(A ’ B ’) Page 3 Random Variables • Definition – Random variable X is a realvalued function of random events in a sample space. • Examples: – T :Timetofailure of a component (Sample space – failures of a component) – N : Number of failed systems out of m systems (Sample space consists of the different combinations of failed and operational systems of the m systems) • Convention: – Capital letters represent the random variables and the lowercase letters denote values of a random variable. Probability P(E) = p Random Variable X(E) = x Page 4 Cumulative Distribution Function (CDF) • Definition – Denoted by F(x) is defined as the probability of a random number X having values less than or equal to x – In terms of set notations: F(x) = P(A) where A ={ a: X(a) ≤ x} – To simply the notation, let P{X ≤ x} = P(A)...
View
Full Document
 Spring '11
 hi
 Probability, Probability theory, probability density function, Cumulative distribution function, CDF

Click to edit the document details