Chapter 3

Chapter 3 - Chapter 3 Discrete Random Variables and...

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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- 1-1 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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Chapter 3 - Chapter 3 Discrete Random Variables and...

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