The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 12 Birthday Problem and Random Variables 5 26 11 Lecture 12 1 Review 5 26 11 Lecture 12 2 5 26 11 Lecture 12 3 5 26 11 Lecture 12 4 5 26 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 5 26 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 5 26 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 5 26 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 5 26 11 Lecture 12 9 Discrete Random Variable 5 26 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 5 26 11 4 16 6 16 Lecture 12 4 16 1 16 11 Probability Histogram 5 26 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 5 26 11 Lecture 12 13 Continuous Random Variable spinner 5 26 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 5 26 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 5 26 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 5 26 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 5 26 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 Three women in the age group are sampled with replacement What is the probability that at least one in the sample are taller than 66 inches What is the probability that more women in the sample are taller than 66 inches see more details on the board 5 26 11 Lecture 12 19
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