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Unformatted text preview: Chapter 3 Discrete Random Variables and Probability Distributions 3.1 Random variable Each outcome of a random experiment can be associated with a (real) number by specifying a rule of association , which is called a random variable Def) A random variable X is a (set) function whose domain is the sample space (S) and whose range is the set of real numbers (R) : X S R Ex 3.1 success and failure experiment S={S,F}, X(S)=1, X(F)=0 Ex 3.2) dial a telephone number using a random number Y=1 if unlisted or 0 if listed Y(5282966)=0, Y(7727350)=1 1 Use X instead of X(A) when there is no confusion Def) Bernoulli random variable a random variable whose only possible values are 0 and 1 Bernoulli experiment repetition of an identical and independent random experiment which has only two outcomes Ex 3.3) Ex 2.3 (two gas stations with 6 pumps each) Define X=total number of pumps in use Y=difference between the number of pumps in use at station 1 and station 2 U=maximum number of pumps in use X((2,3))=5, Y((2,3))=1, U((2,3))=3 Ex 3.4) Ex 2.4(examine battery until first good one) X=number of batteries examined before the experiment terminates X(S)=1, X(FS)=2, X(FFS)=3,, X(FFFFFFS)=7 Ex 3.5) A location (latitude and longitude) in US is selected Y=height above sea level Y((39 50' ,98 35' o o N W ))=1748.26ft 2 Largest possible value of Y=14494ft(Mt. Whitney) Smallest possible value of Y=282(Death valley) How many number of Ys are in the interval bet 282 and 14494  infinite Def) two types of random variables discrete, continuous Discrete rv possible values are finite or infinite sequence 11 correspondence with natural numbers countably many Continuous rv possible values consist of an entire interval uncountably many Ex 3.6 X no of blood tests on married couples until we find the same Rh factor 2,4,6, 3 3.2 Probability Distribution for discrete rv How the total probability 1 is distributed among the various possible values of X values Ex 3.7) no of defectives in six lots 1 2 3 4 5 6 0 2 0 1 2  X=no of defectives in a randomly selected lot p(0)=P(X=0)=3/6 p(1)=P(X=1)=1/6 p(2)=P(X=2)=2/6 Def) Probability distribution, probability function, probability mass function (pmf) of a discrete rv X , ( ) ( ) ( al l : ( ) ) x p x P X x P s S X s x 2200 = = = = Ex 3.8 buy a laptop or desktop X= 1, i f l apt op 0, i f deskt op If 20% buys a laptop, then the pmf will be . 8, i f x=0 ( ) . 2, i f x=1 0, i f x 0, 1 p x = 4 . 8, i f x=0 ( ) . 2, i f x=1 p x = Ex 3.9) Five blood donors (A,B,C,D,E) A, B have O+ Type blood sample until O+ is identified Y=no of typing necessary to identify O+ p(1)=P(Y=1)=2/5=0.4 p(2)=P(Y=2)=(3/5)(2/4)=0.3 p(3)=P(Y=3)=(3/5)(2/4)(2/3)=0.2 p(4)=P(Y=4)=(3/5)(2/4)(1/3)(2/2)=0.1 . 4, i f y=1 . 3, i f y=2 ( ) . 2, i f y=3 . 1, i f y=4 p y = y 1 2 3 4 P(y) .4 .3 .2 .1 5 Def) Suppose pmf p(x) depends on a quantity(...
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 Fall '08
 BUD
 Statistics, Probability

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