ma3160RandomVariable0809B - Random Variable 1 Random...

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Random Variable 1 Random Variable 1.1 Let ( 29 , , P Ω Γ be a probability space. Let X be a function from 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 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 . 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) 0 ( ) 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 , , , = , Γ = P ( 29 , power set of , and ( 29 E n P E = . ( 29 , , P Ω Γ is a probability space. 1
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Consider the function : X R Ω → defined as follows: ( 29 0 X TT = , ( 29 ( 29 1 X TH X HT = = , ( 29 2 X HH = . Observe that X is a discrete random variable. The events { } { } { } 0 1 2 , , , A TT A TH HT A HH = = = clearly form a partition of . And the table of the mass function ( 29 X f x is as follows: ( 29 0 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 step-function W Example A coin, possibly biased, is tossed once. We can take { } T H , = and { } { } { } = Γ , , , T H φ and a possible probability measure R P Γ : is given by ( 29 { } ( 29 { } ( 29 ( 29 1 , 1 , , 0 = - = = = P p T P p H P P φ . Let R X : be given by ( 29 ( 29 0 , 1 = = T X H X . Then X is the simplest non-trivial random variable. Its pmf ( 29 X f x and distribution function (cdf) ( 29 ( 29 X F x P X x = are: 0 1 Probability 1 X p p - and ( 29 0 for 0 1 for 0 1 1 for 1 X x F x p x x < = - < . X is said to have the Bernoulli distribution. W 1.4 A random variable X is called a continuous random variable or to have a continuous distribution if there exists a nonnegative, integrable function ( 29 X f x , defined on the real line, called probability density function (pdf), or just density function of X , such that for every interval like ( ] , b -∞ , where b -∞ < < ∞ , we have 2
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( 29 ( 29 ( 29 { : ( ) } b X P X b P X b f x dx ϖ ϖ -∞ ∈Ω = , where
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