Random Variable

1. Random Variable Tutorial 3STAT1301 Fall 2010 05OCT2010, MB103@HKUBy Joseph Dong

2. A random variable is a function • is called the sample space. It is usually a numbers set, i.e., a subset of or . • A random variable is deterministic. The randomness does not reside in the random variable . The randomness resides in the state space , and is carried by over to the sample space. The Definition

3. What’s the random experiment? • What are the possible outcomes of the random experiment? • What is the state space? • What are the possible values of money you can win? • What is the sample space? • What is the probability measure on the state space? • What is the probability measure on the sample space? • What is the distribution function on the sample space? Handout Problem 1

4. R.E. = Tossing two dice simultaneously to observe 2 digits. • The possible physical outcomes are (1,1), (1,2), ……, (6,6). There are 36 of them. • The State Space is the set of all physical outcomes (states): {(1,1), …, (6,6)}.

5. The possible values of money I can win are 9, -10, and 0. • The Sample Space is the set of all possible numerical outcomes I can win: {9, -10, 0}

6. The (discrete) Probability measure on the state space is a function that evaluates every singleton subset of to and that evaluates every composite subset according to (i) the countable additivity axiom in Kolmogorov’s definition of Probability; and (ii) how it evaluates on the singleton subsets. • For example, will evaluate the composite subset {(1,1), (1,2)} to because • (i) the singleton sets {(1,1)} and {(1,2)} are disjoint and therefore ; and • (ii) .

7. The (discrete) Probability measure on the Sample Space is a function • that evaluates the singleton subsets of , {9}, {-10}, {0} to , , respectively, and • that evaluates every compositesubset according to (i) the countable additivity axiom in Kolmogorov’s definition of Probability; and (ii) how it evaluates on the singleton subsets. • For example, will evaluate the composite subset {9, 0} to because (i) the singleton sets {9} and {0} are disjoint and therefore ; and (ii) and . • Q: How is linked to by ?

8. Choice of R.E. is flexible • R.E. = Tossing two dice simultaneously to observe the sum of the 2 digits produced. OR • R.E. = Tossing two dice simultaneously to observe the value of money you can win based on the sum of the 2 digits produced.

9. Conclusion: Since the 3 spaces above have consistent probability measure, any one can be used as our state space or sample space, depending on your choice. • They are just different representations. • The consistency across the spaces are guaranteed by the defining nature of the random variable between them. • Make sure you use the right probability measure for the sample/state space you work on. • The choice of a good sample space is an art. A good choice of sample space—and accordingly its probability measure—can greatly simplify the solution process.

10. Because of the “deterministic” random variable and the “random” state space, the sample space as the combination of the two is also “random”. • Therefore we usually have each of and endowed with a probability measure, for the depiction of their randomness. • As usual, we use to denote the probability measure on the state space . • We use a subscripted to symbolize the probability measure of the sample space . • must be consistent because is deterministic. A Short Wrap-up vs

11. Going Visual Random Variable/Function

12. Sample Space Illustrated

13. Hint: If you understand our discussion of Problem 1, you should immediately know an example for Problem 6. • Also try to find a different kind of example. Handout Problem 6

14. Cumulative Distribution Function • Probability Density Function • Probability Mass Function • Distribution Function • Density Function • Probability Function Distribution Nomenclature

15. The term distribution function is used in the mathematical literature for never-decreasing functions of which tend to as , and to as . Statisticians currently prefer the term cumulative distribution function, but the adjective “cumulative” is redundant. • A density function is a non-negative function whose integral, extended over the entire -axis, is unity. • The integral from to of any density function is a distribution function. • —William Feller: An Introduction to Probability Theory and Its Applications (1950) Volume I, page 179: “Note on Terminology.”

16. Handout Problem 3 • Hint: • Handout Problem 4 • Hint: Just routine calculation. Handout Problem 3 and 4

17. We only define distribution function from to . Therefore, strictly speaking, only real-valued random variables can have a distribution function defined for its sample space. • A distribution function is just an alternative way besides the probability measure to depict randomness. • Relationship: • Still with Problem 1, what’s the distribution function on the sample space {-10, 0, 9}? Distribution Function

18. Draw a graph for each question to show the random variable, the state space, the sample space, and the probability mass function on the sample space. Handout Problem 2

19. This time the state space is the interval on the real line. • Try to use use the state space and the sample space as the two ordinates of a Cartesian plane, and draw the graph of the random variable on that coordinated plane. Handout Problem 5

20. Hint: Try to understand “Expectation is the coordinate of the center of mass” Handout Problem 7