{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Ch2 - Statistics 427 Introduction to Probability and...

Info icon This preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
Statistics 427 Introduction to Probability and Statistics I Instructor: Prof. Mark Berliner Email: [email protected] Office: CH 205B (Cockins Hall) Phone: 614.292.0291 http://www.stat.osu.edu/~mb Course information available at:
Image of page 1

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

View Full Document Right Arrow Icon
Chapter 2 Intro to basic prob. theory Point: Mathematical models and analyses for treating uncertainty Key topics: 1. Formulation 2. Set theory and Venn diagrams 3. Probability measures (PM) 4. Basic properties and examples of PM 5. Conditional prob. and Bayes‟ Theorem
Image of page 2
3 1. Formulation Outcome: a possible result of the random phenomenon studied. Experiment: any process leading to an uncertain (= “random”) outcome. Sample space S: set of all possible outcomes. Event: a subset of S. (Notation: A, B, etc.) Ex) Expt: observe temperature at 5pm at Columbus airport. S: all real numbers (OK to make S too big) Event: temp is below freezing A={x | x < 32F}
Image of page 3

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

View Full Document Right Arrow Icon
4 Ex) Expt: flip a coin twice. (remark: silly examples just for rapid illustration!) S = {(h,h), (h,t), (t,h), (t,t)} or S 2 = {0 h‟s, 1 h, 2 h‟s} (remark: sample spaces aren‟t unique. “Art” in choosing S to be useful and efficient. ) Events (using S): A: “one h” …. {(h,t), (t,h)} B: “match” …. {(h,h), (t,t)}
Image of page 4
2. Events, Sets, and Venn Diagrams Box represents S A B
Image of page 5

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

View Full Document Right Arrow Icon
Main operations Intersection: A I B Union: A U B
Image of page 6
Complement: A A, B mutually exclusive or disjoint A A A B
Image of page 7

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

View Full Document Right Arrow Icon
8 3. Probability measures A probability measure (PM) is a function that assigns numbers between 0 and 1 to events. P: events [0,1] Note: “regular” functions, say f. y = f(x) where x and y are reals PM: p = P(A) where p is a real number in [0, 1] and A is a subset of S. Graphs of f ‟s are useful, but we can‟t generally graph P . Events are not typically “well - ordered”
Image of page 8
9 Rules (“axioms”) for all PM (p. 51) 1. For any event A, P(A) > 0 2. P(S) = 1 3. Additivity. For any collection of disjoint sets: A 1 , A 2 , A 3 , …. P(A 1 U A 2 U A 3 U …. ) = P(A 1 ) + P(A 2 ) + …. i.e., for disjoint events, Prob(union) = sum of individual prob‟s
Image of page 9

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

View Full Document Right Arrow Icon
10 4. Basic properties of PM P(A ) = 1 P(A) Since A and A are disjoint and their union is S. Remark: sometimes easier to find P(A ) than to find P(A) directly. P( f ) = 0. (“empty set”, f ) If A is a subset of B, P(A) < P(B) General addition rules P(A U B) = P(A) + P(B) - P(A I B)
Image of page 10
P(A U B) = P(A) + P(B) - P(A I B) A B
Image of page 11

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

View Full Document Right Arrow Icon
12 Types of sample spaces 1. Discrete (or countable) S = {E 1 , E 2 , E 3 , …. } 2. Continuous S is some interval of real numbers Cover discrete case first: Fundamental fact for discrete S (p. 56): For any event A, P(A) = S E i in A P({E i }) (Continuous case: integrate to find prob‟s )
Image of page 12
13 Ex) Geiger counter. Record number of particles hitting plate in some fixed time period.
Image of page 13

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

View Full Document Right Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

{[ 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