Lec 1D - Page 1 Probability of Independent Events •...

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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 real-valued function of random events in a sample space. • Examples: – T :Time-to-failure 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)...
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Lec 1D - Page 1 Probability of Independent Events •...

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