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xyFeb6Lec9 - Two types of random variables A discrete...

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Unformatted text preview: Two types of random variables A discrete random variable has a finite number of possible values or an infinite sequence of countable real numbers. X: number of hits when trying 20 free throws. X: number of customers who arrive at the bank from 8:30 9:30AM Mon-Fri. E.g. Binomial, Poisson ... A con*nuous random variable takes all values in an interval of real numbers. X: the Qme it takes for a bulb to burn out. The values are not countable. Lecture 8 2/7/12 1 Discrete Random Variable 2/7/12 Lecture 8 2 Example 1: Flip a coin 4 Qmes Find the probability distribuQon of the random variable describing the number of heads that turn up when a coin is flipped four Qmes. Solu*on 1/16 2/7/12 4/16 6/16 Lecture 8 4/16 1/16 3 Probability Histogram 2/7/12 Lecture 8 4 Example 2: randomly sampling and tesQng 10 items from a shipment Suppose it's known that 5% of all items do not conform to quality standards. A = at most one of the sampled items fails the test B = none of the sampled items passes the test C = exactly one item fails to meet standards D = at least one fails to meet standards Define r.v. X = # of items that fail the test, so A -> X 1 B -> (1) X = 0 or (2) X = 10? C -> X = 1 D -> X 1 2/7/12 Lecture 8 5 Answers X ~ Binomial(10, .05), therefore P(A) = P(0) + P(1) = .914 P(B) = P(10) = .000 P(C) = P(1) = .315 P(D) = 1 P(0) = 1 - .599 = .401 2/7/12 Lecture 8 6 ConQnuous Random Variable A conQnuous random variable X takes all values in an interval of numbers. e.g. life Qme of a regular bulb. Not countable The probability distribuQon of a conQnuous r.v. X is described by a density curve. What is a density curve? 2/7/12 Lecture 8 7 ConQnuous DistribuQon The probability of any event is the area under the density curve and above the values of X that make up the event. 2/7/12 Lecture 8 8 ConQnuous DistribuQon The probability model for a conQnuous random variable assigns probabiliQes to intervals of outcomes rather than to individual outcomes. In fact, all conQnuous probability distribuQons assign probability 0 to every individual outcome. The spinner Examples: Normal distribuQons, ExponenQal DistribuQons, Uniform DistribuQons, etc. Self-Reading: example 5.10 on Pg 216-217. 2/7/12 Lecture 8 9 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. Suppose their heights are independent. What is the probability that both of them are taller than 66 inches? Define X = an American woman's height, X ~N(64.3, 2.4). For only one woman, P(X > 66inches) = P(both are taller than 66 in.) = P(X1> 66 in.)*P(X2>66 in.) = 2/7/12 Lecture 8 10 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. Suppose their heights are independent. What is the probability that both of them are taller than 66 inches? Define X = an American woman's height, X ~N(64.3, 2.4). For only one woman, P(X > 66inches) = P(both are taller than 66 in.) = P(X1> 66 in.)*P(X2>66 in.) = 2/7/12 Lecture 9 11 Review: Mean (Expected Value) of a r.v. Mean of a conQnuous r.v. Let f(x) be the density funcQon for a conQnuous random variable X, then the mean of X is: X = x f ( x)dx - Mean of a discrete r.v. Let p(x) be the mass funcQon for a conQnuous random variable X, then the mean of X is: 2/7/12 X = x p(x) Lecture 9 12 Examples--Means of some r.v.s ConQnuous distribuQons Normal (,) -- ExponenQal () -- 1/ Uniform (a, b) -- (a+b)/2 Discrete distribuQons Binomial (n, ) -- n Poisson () -- 2/7/12 Lecture 9 13 Free-throws A BoilerMaker basketball player is a 80% free-throw shooter. Suppose he will shoot 5 free-throws during each pracQce. X: number of hits he makes during the pracQce. Find the mean of X. = n* = 5 * 80% = 4 2/7/12 Lecture 9 14 Review: Variance of a r.v. Variance for conQnuous r.v.s Let f(x) be the density funcQon for a conQnuous random variable X, then the variance of X is: Variance for discrete r.v.s 2 X = 2 X (x - X ) - 2 f ( x)dx Let p(x) be the mass funcQon for a conQnuous random variable X, then the variance of X is: - Standard deviaQon of X, is square root of the X 2 variance X 2/7/12 Lecture 9 15 = (x - X ) p ( x ) 2 Examples--Variances of certain r.v.s ConQnuous distribuQons Normal (,) -- 2 ExponenQal () -- 1/ 2 Uniform (a, b) -- (ba)2/12 Discrete distribuQons Binomial (n, ) -- n(1) Poisson () -- 2/7/12 Lecture 9 16 Car Sales The total number of cars to be sold next week is described by the following probability distribuQon x p(x) 0 1 .05 .15 2 .35 3 .25 4 .20 Determine the expected value and standard deviaQon of X, the number of cars sold. X = xi p( xi ) = 0(0.05) + 1(0.15) + 2(0.35) + 3(0.25) + 4(0.20) = 2.40 i =1 5 5 X = ( xi - 2.4) 2 p( xi ) = (0 - 2.4) 2 (.05) + (1 - 2.4) 2 (.15) i =1 2 + (2 - 2.4) 2 (.35) + (3 - 2.4) 2 (.25) + (4 - 2.4) 2 (.20) = 1.24 2/7/12 X = 1.24 = 1.11 Lecture 9 17 Independent r.v.s Ex.41 (Pg 223), Part(c) Are X and Y independent? y 10 15 20 5 .20 .15 .05 x 6 .10 .15 .10 7 .10 .10 .05 Answer: No, because for X=5 and Y = 10, P(X=5 and Y=10) = 0.20, while P(X=5)P(Y=10) = .4*0.4 = 0.16, i.e. P(X=5 and Y=10)P(X=5)P(Y=10). 2/7/12 Lecture 9 X and Y are said to be independent if the events X < a and Y < b are independent for all possible combinaQons of real numbers a and b, i.e. P(X< a and Y < b) = P(X< a)P(Y< b). For Discrete r.v. ONLY: We can use "=" to replace "<". 18 Awer Class... Review Sec 5.4 Read Sec 5.5 and 5.6 Hw#4, due by 5pm next Monday. 2/7/12 Lecture 9 19 ...
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