Probability_lecture

# Probability_lecture - Probability for Math 441 November 2...

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Unformatted text preview: Probability for Math 441 November 2, 2010 Outline Definition of Probability Random variables Expectation for a finite sample space Variance and standard deviation c.d.f and p.d.f Expectation (revisited) The normal variable Outline Definition of Probability Random variables Expectation for a finite sample space Variance and standard deviation c.d.f and p.d.f Expectation (revisited) The normal variable Want to define the notion of probability carefully. Want to define the notion of probability carefully. First we need a set of outcomes of an experiment, denoted Ω . Want to define the notion of probability carefully. First we need a set of outcomes of an experiment, denoted Ω . For example, Ω is the set of outcomes of flipping 3 coins or the possible prices of a stock at a given time. Want to define the notion of probability carefully. First we need a set of outcomes of an experiment, denoted Ω . For example, Ω is the set of outcomes of flipping 3 coins or the possible prices of a stock at a given time. In the first case, Ω = { HHH , HHT , ... } . And Ω is finite with 8 elements. Want to define the notion of probability carefully. First we need a set of outcomes of an experiment, denoted Ω . For example, Ω is the set of outcomes of flipping 3 coins or the possible prices of a stock at a given time. In the first case, Ω = { HHH , HHT , ... } . And Ω is finite with 8 elements. In the second case, Ω is infinite and is the set of positive real numbers. Given a set of outcomes Ω (also called the sample space), we can define a probability on these outcomes. Definition Let Ω be the sample space of an experiment. To each event A ⊂ Ω , we assign a number p ( A ) , called the probability of A. Given a set of outcomes Ω (also called the sample space), we can define a probability on these outcomes. Definition Let Ω be the sample space of an experiment. To each event A ⊂ Ω , we assign a number p ( A ) , called the probability of A. The probability function p satisfies some conditions: 1. ≤ p ( A ) ≤ 1 for each subset A ⊂ Ω . 2. p (Ω) = 1 and p ( ∅ ) = . 3. For A , B ⊂ Ω , we have p ( A ∪ B ) = p ( A ) + p ( B ) if A ∩ B = ∅ . Given a set of outcomes Ω (also called the sample space), we can define a probability on these outcomes. Definition Let Ω be the sample space of an experiment. To each event A ⊂ Ω , we assign a number p ( A ) , called the probability of A. The probability function p satisfies some conditions: 1. ≤ p ( A ) ≤ 1 for each subset A ⊂ Ω . 2. p (Ω) = 1 and p ( ∅ ) = . 3. For A , B ⊂ Ω , we have p ( A ∪ B ) = p ( A ) + p ( B ) if A ∩ B = ∅ . Roughly speaking, p ( A ) measures the percent of occurrences where the outcome of the experiment lies in A when the experiment is performed more and more times. Outline Definition of Probability Random variables Expectation for a finite sample space Variance and standard deviation c.d.f and p.d.f Expectation (revisited) The normal variable Next we discuss random variables. Next we discuss random variables....
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Probability_lecture - Probability for Math 441 November 2...

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