lect3a - 3 Random Variables Let F P be a probability model...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '09
  • prof
  • Probability distribution, Probability theory, probability density function, Cumulative distribution function, lim FX

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern