lect3 - 3. Random Variables Let (, F, P) be a probability...

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1 3. Random Variables Let ( , F , P ) be a probability model for an experiment, and X a function that maps every to a unique point the set of real numbers. Since the outcome is not certain, so is the value Thus if B is some subset of R , we may want to determine the probability of ”. To determine this probability, we can look at the set that contains all that maps into B under the function X . , ξ , R x ξ . ) ( x X = ξ B X ) ( ξ = ) ( 1 B X A ξ ξ R ) ( ξ X x A B Fig. 3.1 PILLAI
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2 Obviously, if the set also belongs to the associated field F , then it is an event and the probability of A is well defined; in that case we can say However, may not always belong to F for all B , thus creating difficulties. The notion of random variable (r.v) makes sure that the inverse mapping always results in an event so that we are able to determine the probability for any Random Variable (r.v) : A finite single valued function that maps the set of all experimental outcomes into the set of real numbers R is said to be a r.v, if the set is an event for every x in R . ) ( 1 B X A = )). ( ( " ) ( " event the of y Probabilit 1 B X P B X = ξ (3-1) ) ( 1 B X . R B ) ( X {} ) ( | x X ξ ξ ) ( F PILLAI
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3 Alternatively X is said to be a r.v, if where B represents semi-definite intervals of the form and all other sets that can be constructed from these sets by performing the set operations of union, intersection and negation any number of times. The Borel collection B of such subsets of R is the smallest σ -field of subsets of R that includes all semi-infinite intervals of the above form. Thus if X is a r.v, then is an event for every x . What about Are they also events ? In fact with since and are events, is an event and hence is also an event. } { a x < −∞ a b > {} { } ? , a X b X a = < b X } { b X a b X a X < = > a X a X c > = { } ) ( | x X x X = ξ ξ F B X ) ( 1 } { a X (3-2) PILLAI
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4 Thus, is an event for every n . Consequently is also an event. All events have well defined probability. Thus the probability of the event must depend on x . Denote The role of the subscript X in (3-4) is only to identify the actual r.v. is said to the Probability Distribution Function (PDF) associated with the r.v X . < 1 a X n a = = = < 1 } { 1 n a X a X n a {} ) ( | x X ξ ξ . 0 ) ( ) ( | = x F x X P X ξ ξ (3-4) ) ( x F X (3-3) PILLAI
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5 Distribution Function : Note that a distribution function g ( x ) is nondecreasing, right-continuous and satisfies i.e., if g ( x ) is a distribution function, then (i) (ii) if then and (iii) for all x . We need to show that defined in (3-4) satisfies all properties in (3-6). In fact, for any r.v X , , 0 ) ( , 1 ) ( = −∞ = +∞ g g , 0 ) ( , 1 ) ( = −∞ = +∞ g g , 2 1 x x < ), ( ) ( 2 1 x g x g ), ( ) ( x g x g = + (3-6) ) ( x F X (3-5) PILLAI
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6 {} 1 ) ( ) ( | ) ( = = +∞ = +∞ P X P F X ξ ξ . 0 ) ( ) ( | ) ( = = −∞ = −∞ φ ξ ξ P X P F X (i) and (ii) If then the subset Consequently the event since implies As a result implying that the probability distribution function is nonnegative and monotone nondecreasing.
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This note was uploaded on 10/01/2009 for the course ECE 2521 taught by Professor Jacobs during the Fall '09 term at Pittsburgh.

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lect3 - 3. Random Variables Let (, F, P) be a probability...

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