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Unformatted text preview: Lecture 2 2.1 Some probability distributions. Let us recall some common distributions on the real line that will be used often in this class. We will deal with two types of distributions: 1. Discrete 2. Continuous Discrete distributions. Suppose that a set X consists of a countable or finite number of points, . X = { a 1 , a 2 , a 3 , ···} Then a probability distribution on X can be defined via a function p ( x ) on X with the following properties: 1. ∀ p ( a i ) ∀ 1 , 2. ∗ i =1 p ( a i ) = 1 . p ( x ) is called the probability function. If X is a random variable with distribution then p ( a i ) = ( a i ) is a probability that X takes value a i . Given a function : X ≈ , the expectation of ( X ) is defined by ∗ X ( X ) = ( a i ) p ( a i ) i =1 (Absolutely) continuous distributions. Continuous distribution on is defined via a probability density function ∗ (p.d.f.) p ( x ) on such that p ( X ) → and p ( X ) dx = 1 . If a random vari −∗ able X has distribution then the chance/probability that X takes a value in the 3 4 LECTURE 2. interval [ a, b ] is given by b ( X ⊆ [ a, b ]) = p ( x ) dx. a Clearly, in this case for any a ⊆ we have ( X = a ) = 0. Given a function , the expectation of ( X ) is defined by : X ≈ ∗ ( X ) = ( x ) p ( x ) dx. −∗ Notation. The fact that a random variable X has distribution will be denoted by X ∩ . Example 1. Normal (Gaussian) Distribution N ( ϕ, δ 2 ) with mean ϕ and variance δ 2 is a continuous distribution on with probability density function: 1 ( x −...
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
 Spring '09
 DmitryPanchenko
 Statistics, Probability

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