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Unformatted text preview: Random Variable 1 Random Variable 1.1 Let ( 29 , , P Ω Γ be a probability space. Let X be a function from Ω to R , that is, : X R Ω → , where R is the set of real numbers. X is said to be a random variable if for every real number a , { } Γ ∈ ≤ Ω ∈ a X ) ( : ϖ ϖ , therefore, for a random variable X , { } ( 29 : ( ) ( ) is defined P X a P X a ϖ ϖ ∈ Ω ≤ ≡ ≤ . 1.2 Let : X R Ω → be a random variable. The cumulative distribution function (cdf) or just a distribution of X is defined for any real number x by { } ( 29 ( ) : ( ) ( ) X F x P X x P X x ϖ ϖ = ∈Ω ≤ ≡ ≤ . 1.3 X is said to be a discrete random variable if the range of X is discrete (the range of X contains finitely many or countably infinitely many real numbers). Let : X R Ω → be a discrete random variable. Suppose ( 29 { } 1 , , , i X x x Ω = L L . Let ( 29 { } : i i A X x ϖ ϖ = = , 1,2, i = L . Observe that ( 29 { } : i i A X x ϖ ϖ = = , 1,2, i = L is a sequence of events, which form a partition of the sample space Ω . The function ( ) X f x , called the probability mass function (pmf) , or just mass function, of X is defined as follows: ( 29 { } ( 29 ( 29 ( 29 { } { } 1 2 1 : = if ,where , , , , ( ) 0 if , , , , i i x i i i X i P X x P A x x x X x x x f x x x x ϖ ϖ ∈Ω = = ∈ Ω = = ∉ L L L L L . ( ) X f x satisfies: (a) ( ) 1 X f x ≤ ≤ , (b) { } 1 2 , , ( ) 1 X x x x f x ∈ = ∑ L , (c) ( 29 ( 29 { } ( 29 ( 29 { } 1 2 , , : X x A x x P X A P X A f x ϖ ϖ ∈ ∩ ∈ ≡ ∈Ω ∈ = ∑ L , where A R ⊂ . The probability model of a discrete random variable X is a list of the distinct values x i of X together with their associated probabilities ( 29 { } ( 29 ( 29 ( ) : X i i i f x P X x P X x ϖ ϖ = ∈Ω = ≡ = : ( 29 ( 29 ( 29 ( 29 1 2 3 1 2 3 X X X X X x x x f x f x f x f x L L Example# If a coin is tossed twice, the number of heads, which turn up, can be 0, 1 or 2 according to the outcome of the experiment. Sample space { } HH HT TH TT , , , = Ω , Γ = P ( 29 Ω , power set of Ω , and ( 29 E n P E = Ω . ( 29 , , P Ω Γ is a probability space. 1 Consider the function : X R Ω → defined as follows: ( 29 X TT = , ( 29 ( 29 1 X TH X HT = = , ( 29 2 X HH = . Observe that X is a discrete random variable. The events { } { } { } 1 2 , , , A TT A TH HT A HH = = = clearly form a partition of Ω . And the table of the mass function ( 29 X f x is as follows: ( 29 1 2 1 1 1 4 2 4 X X f x . W Example The cdf of X in example# can be represented by a stepfunction W Example A coin, possibly biased, is tossed once. We can take { } T H , = Ω and { }{ } { } Ω = Γ , , , T H φ and a possible probability measure R P → Γ : is given by ( 29 { } ( 29 { } ( 29 ( 29 1 , 1 , , = Ω = = = P p T P p H P P φ ....
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This note was uploaded on 01/11/2011 for the course MA 3160 taught by Professor Cwwoo during the Spring '07 term at City University of Hong Kong.
 Spring '07
 CWWoo
 Real Numbers, Probability

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