Probability Theory and Simulation Methods

Transcription

1 Feb 28th, 2018 Lecture 10: Random variables

2 Countdown to midterm (March 21st): 28 days Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 5, 6, 7: Random variables Week 9 Chapters 8, 9: Bivariate and multivariate distributions Week 10 Chapter 10: Expectations and variances Week 11 Chapter 11: Limit theorems Week 12 Chapters 12, 13: Selected topics

3 Order Discrete random variables (Chap 4) Continuous random variables (Chap 6) Special discrete distributions (Chap 5) Special continuous distributions (Chap 7)

4 Chapter 4: Discrete random variables 4.1 Random variables 4.3 Discrete random variables 4.4 Expectations of discrete random variables 4.5 Variances and moments of discrete random variables 4.2 Distribution functions

5 Random variable Definition Let S be the sample space of an experiment. A real-valued function X : S R is called a random variable of the experiment.

6 Example: tossing a coin two times Experiment: Toss a fair coins 2 times Let X be the number of heads X is a random variable What are all possible values of X? What is the probability that X = 1?

7 A random variable is a function... X is a function that maps S = {HH, HT, TH, TT} to {0, 1, 2} such that HH 2 HT 1 TH 1 TT 0

8 ... and its value is random X probability Remark: the probability on the sample space induces a new probability on R

9 Example: Brownian motion Experiment: let a pollen grain move in water, starting at a given position, for 1s Einstein s question: the trajectory of the pollen grain An outcome: a record of the locations and velocities of all the molecules in the system over time The sample space: the set of all possible outcomes

10 Einstein s approach Describes the increment of particle positions in unrestricted one dimensional domain as a random variable Computes the probability law of the probability law of the location of the particle after time t

11 ( Enhanced ) definition of random variables Definition Let S be the sample space of an experiment [that is so complicated and obscure that we can not learn anything from it]. A random variable of the experiment is just a real-valued function X : S R [for which people try to forget about its domain of definition].

12 Notations When we write {X = 2}, we are (implicitly) referring to the set {s S : X(s) = 2} which is an event of the experiment (a subset of S). When we write {X (0, 1)}, we are (implicitly) referring to the event {s S : X(s) (0, 1)}.

13 Question X probability Question: If we don t know the probability on the sample space (S), how can we compute the probability induced on X Answer: We can t. But at least X is quantifiable and usually observable.

14 Bayesian statistics Bayesian learning: Start with some prior belief on how likely each outcome of X happen With new information comes in (events E 1, E 2,..., happen), we update the belief about the likelihood of each outcome

15 Discrete random variables

16 Discrete random variable Definition A random variables X is discrete if the set of all possible values of X is finite is countably infinite Note: A set A is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers, i.e, we can index the element of A as a sequence A = {x 1, x 2,..., x n,...}

17 Discrete random variable A random variable X is described by its probability mass function

18 Discrete random variable: finite case x p(x)

19 Exercise Problem Let p be a function defined as follows ( ) 1 2 x 2 3 if x = 1, 2, 3,..., p(x) = 0 elsewhere Prove that p is a probability mass function.

20 Read the pmf table

21 A game of chance Let s assume we have a biased coin that turns head 60% of the time. Consider two scenarios Scenario 1: I toss the coin; if the outcome is tail, you win $7; if it is head, you lose $5 Scenario 2: I toss the coin; if the outcome is tail, you win $6; if it is head, you lose $4 Which scenario is a fair game?

22 Expectation Example: Let X be the money you get by playing one game in Scenario 1 x 7-5 p(x) then E(X) = 7 (0.4) + ( 5) (0.6) = 2.