Random Variables. Marina Santini. Department of Linguistics and Philology Uppsala University, Uppsala, Sweden

Transcription

1 Random Variables Marina Santini Department of Linguistics and Philology Uppsala University, Uppsala, Sweden Spring 2016

2 Acknowledgements Wikipedia Tamhane A. and Dunlop D. (2000). Statistics and Data Analysis. Prentice Hall. Ross S. (2014). A first course in Probability, Pearson, 9th edition. Kracht M. (2005). Introduction to Probability Theory and Statistics for Linguistics. Dpt of Linguistics, UCLA Mollevan J. (2008). Introduction to Probability Theory and Statistics. Introduction to STAT 414/415: PennState Uni 2

3 Required Reading for this lecture Handouts (see course website: 4LTechnologists.html. 3

4 Outline of the section Random variables Descrete and continuous Probability mass function (pmf) Probability density function (pdf) 4

5 Intuitively, a random variable Intuitively, we can think of a random variable as a numerical measurement of outcomes. A random variable is a rule (i.e., a function) that associates numbers to outcomes. 5

6 (repetition: functions) A function is a rule that associates members of two sets. The first set is called the domain and the second set is called the target or codomain. This rule has to be such that an element of the domain should not be associated to more than one element of the codomain. Functions are usually described using the following notation f : A B where f is the symbol identifying the function, A is the domain and B is the target. Another example, h : R R tells us that h is a function whose inputs are real numbers and whose outputs are also real numbers. The function h(x) = (2)(x)+4 would satisfy that description. 6

7 Random Variable A random variable associates a unique numerical value with each outcome in the sample space. Formally, a random variable is a single real-valued function defined over a sample space. Often, we denote a random variable by a capital letter (eg. X or Y) and a particular value taken by a random variable by the corresponding lower case (x or y). 7

8 Random Variable: Examples Ex 1: Run a laboratory test. X = 1, if the result is positive X = 0, if the result is negative Ex 2: Toss two dice. X = sum of the numbers on the dice. Ex 3: Observe how long a transistor lasts. X=lifetime the transistor. 8

9 Definition We can also say that: a random variable X : Ω R is a measurable function from the set of possible outcomes Ω to some set R. This definition requires Ω to be a probability space and R to be a measurable space. 9

10 Random Variables: Discrete and Continuous Discrete random variables take DISTINCT, SEPARABLE values: 1 = Heads X = 0 = Tails Continuous random variable can take any value within a range: Examples: A person's height: could be any value (within the range of human heights), not just certain fixed heights, Time in a race: you could even measure it to fractions of a second, etc. 10

11 Another ex: English words The words of English have a certain length probability (eg. short words are more frequent than longer words) the function X that assigns to each word a length is a random variable for the discrete space over all the words of English. 11

12 Discrete Random Variables The possible values of a discrete rv can be listed as x 1, x 2 etc. Suppose we can calulate P(X = x) for every value x that X can take. 12

13 Probability Mass Function (p.m.f) The collection of these probabilities can be viewed as a function of x: We denote them by f(x) = P(X = x) for each x The function f(x) is called the probability mass function (p.m.f.) 13

14 p.m.f A probability mass function maps real numbers to probabilities for any value x in the range of the random variable X. f(x) =P(X = x) 14

15 Continous Random Variables A random variable X is continuous if it can assume any value from one or more interval of real numbers. 15

16 Probability Density Function (p.d.f) We need to find the probability that X falls in some interval (a, b), ie, we need to find P(a < X < b). We do that using a probability density function ("p.d.f."). 16

17 End of the section 17