MATH/STAT 395. Introduction to Probability Models. Jan 7, 2013

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1 MATH/STAT 395 Introduction to Probability Models Jan 7, 2013

2 1.0 Random Variables Definition: A random variable X is a measurable function from the sample space Ω to the real line R. X : Ω R Ω is the set of outcomes corresponding to an experiment. ω is one possible outcome in the outcome set. The word measurable has a precise meaning which we will not deal with here.

3 2.0 Tay-Sachs Disease from Prof. Guttorp Tay-Sachs is a rare but fatal genetic disease. It is known that if a couple both carry the defective gene, then their children have probability 0.25 of being born with it. Consider an experiment that studies such a couple with 3 children. The investigator notes whether each child is affected ( a ) or disease-free ( f ). 1. What is the sample space Ω corresponding to this experiment? What is p(ω), the probability assigned to each outcome ω? 2. Let X denote the number of children who have the defective gene. Write X as a function that maps Ω to the real line. What are the probabilities assigned to each possible value of X?

4 2.1 Bernoulli Trials Suppose the experiment consists of performing n independent Bernoulli trials. trials with outcomes S or success and F or failure; The success probability is p for each trial. the outcome of one trial has no physical influence over another. Ω contains 2 n outcomes. Each outcome ω is a sequence of n S s and F s. The probability associated with each ω is: p(ω) = p no. of S s (1 p) no. of F s

5 2.2 Binomial Probabilities For any outcome ω, let X denote the number of S s that appear in ω. The probabilities for the various possible values of X are: P(X = x) = P(ω that have x S s) = ( ) n p x (1 p) n x, x = 0, 1,..., n. x X is called a binomial random variable.

6 2.3 Tay-Sachs re-visited In the Tay-Sachs example which is more likely? At least one affected child in a two-child family or at least 2 affected children in a four-child family?

7 3.0 Poisson Probabilities Consider an experiment where we record the number of automobile accidents in a week in a large city. Then the sample space Ω is the set of non-negative integers {0, 1, 2, 3,... } If the average number of automobile accidents per week is 5.7, then it is usually not a bad approximation to calculate the probability of (say) 4 accidents in a given week as: More generally: p(4) = e 5.7 /4! = p(ω) = 5.7 ω e 5.7 /ω!, ω = 0, 1, 2...

8 3.1 Poisson Random Variable If we are just interested in the value of the number chosen at random, then there is no need to define a random variable. Why? But we could define X(ω) = ω say. The probabilities for the various possible values are: P(X = x) = p(x), = λ x e λ /x!, x = 0, 1, 2,... where λ > 0 is the rate of occurrence of the event in the time period. X is called a Poisson random variable.

9 4.0 The Spinner Experiment Now consider an experiment where a well-balanced spinner is spun to generate a number at random from [0, 1) The outcome set is Ω = [0, 1).

10 4.1 Uniform Probability Model For any a and b in [0, 1) with a < b the probability that the spinner comes to rest at an ω in the interval [a, b) is given by its length b a. According to this model, all singletons (like 0.5 say) have probability 0. Why? What does this mean? Therefore, according to this model, for any a and b in [0, 1) with a < b, the four intervals [a, b], [a, b), (a, b], and (a, b) all have probability b a. Note that the probability b a can be written as f(ω)dω, where f(ω) = 1 for all ω [0, 1). b a

11 4.2 Uniform Random Variable Again, there is no need to define a random variable, since the outcomes themselves are numbers. It is economical (notationally) though to define the random variable X(ω) = ω. X is called a uniform random variable. There might also be other random variables of interest, such as, Y (ω) = ln(ω).

12 5.0 Probability Models Fact: In a discrete probability model Ω is finite or countable Probabilities are calculated using a probability mass function p(ω) which assigns probabilities to each outcome ω: p(ω) 0 ω Ω and ω p(ω) = 1 A random variable defined in this setting is called a discrete random variable.

13 5.0 Probability Models Fact: In an absolutely continuous probability model in R Ω is an interval The probability that an outcome lies in any sub-interval I can be calculated as P(ω I) = f(u)du, where f() is called a probability density function and satisfies: f(ω) 0 for each ω Ω and f(ω)dω = 1. Ω A random variable defined in this setting is called a continuous random variable. I

Lecture 3. Discrete Random Variables

Math 408 - Mathematical Statistics Lecture 3. Discrete Random Variables January 23, 2013 Konstantin Zuev (USC) Math 408, Lecture 3 January 23, 2013 1 / 14 Agenda Random Variable: Motivation and Definition