The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 12 Birthday Problem and Random Variables 3 1 11 Lecture 12 1 Review 3 1 11 Lecture 12 2 3 1 11 Lecture 12 3 3 1 11 Lecture 12 4 3 1 11 Lecture 12 5 Birthday Problem In a classroom of 45 people what is the probability that at least two people have the same birthday Event A at least two people have the same birthday out of the 45 people AC every person has a different birthday out of the 45 people P A 1 P AC see the board 3 1 11 Lecture 12 6 Free throws A TarHeel basketball player is a 80 free throw shooter Suppose he will shoot 20 free throws during each practice Which is more likely to make 5 out of 20 or 18 out of 20 How many free throws he makes on average during practice 3 1 11 Lecture 12 7 Random Variables Experiment A TarHeel basketball player shoots 20 free throws during his practice X number of hits A random variable is a variable whose value is a numerical outcome of a random experiment 3 1 11 Lecture 12 8 Two types of random variables A discrete random variable has a finite number of possible values X number of hits when trying 20 free throws Possible values for X 0 1 20 A continuous random variable takes values in an interval X the time it takes for a bulb to burn out Possible values are not countable 3 1 11 Lecture 12 9 Discrete Random Variable 3 1 11 Lecture 12 10 Flip a fair coin 4 times Find the probability distribution of the random variable describing the number of heads that turn up when a fair coin is flipped 4 times Solution 1 16 3 1 11 4 16 6 16 Lecture 12 4 16 1 16 11 Probability Histogram 3 1 11 Lecture 12 12 Questions What is the connection with histograms we talked about in Chapter 1 Are the two problems similar toss a coin 4 times and shoot 20 free throws Yes or no the free throw problem is equivalent to tossing a biased coin 20 times each with P H 0 8 3 1 11 Lecture 12 13 Continuous Random Variable spinner 3 1 11 Lecture 12 14 Continuous Random Variable A continuous random variable X takes all possible values in an interval Not countable The probability distribution of a continuous r v X is described by a density curve What is a density curve 3 1 11 Lecture 12 15 Ex Spinner continued P point to 1 4 0 why P greater than 5 8 1 5 8 3 8 P between 2 9 and 7 8 7 8 2 9 P falling in x x 1 4 1 4 for any x greater than 0 and less than 3 4 3 1 11 Lecture 12 16 Continuous Distribution The probability of any event is the area under the density curve and above the values of X that make up the event 3 1 11 Lecture 12 17 Continuous Distribution The probability model for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes In fact all continuous probability distributions assign probability 0 to every individual outcome The spinner Normal distributions are continuous probability distributions 3 1 11 Lecture 12 18 Women Height The height of American women aged 18 24 is approximately normally distributed with mean 64 3 inches and s d 2 4 inches Two women in the age group are randomly selected What is the probability that both of them are taller than 66 inches see more details on the board 3 1 11 Lecture 12 19
View Full Document
Unlocking...