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Unformatted text preview: 0.1 Exercises for Chapter 4 1. In a gasoline station there are two selfservice pumps and two fullservice pumps. Let X denote the number of selfservice pumps used at a particular time and Y the number of fullservice pumps in use at that time. The joint pmf p ( x,y ) of ( X,Y ) appears in the next table. y 1 2 .10 .04 .02 x 1 .08 .20 .06 2 .06 .14 .30 a) Find the probability P ( X 1 ,Y 1). b) Compute the marginal pmf of X and Y . c) Compute the mean value and variance of X and Y . d) Compute the covariance and correlation between X and Y . 2. Suppose that X and Y are discrete random variables taking values 1, 0, and 1, and that their joint pmf is given by X p ( x,y )1 11 1/9 2/9 Y 1/9 1/9 1/9 1 2/9 1/9 a) Find the marginal pmf of X . b) Are X and Y independent? Why? c) Find E ( X ). d) Find Cov ( X,Y ). e) Find the probability P ( X 0 and Y 0). f) Find E ( max { X,Y } ). 1 3. The following table shows the joint probability distribution of X , the amount of drug administered to a randomly selected laboratory rat, and Y , the number of tumors present on the rat. y p ( x,y ) 1 2 0.0 mg/kg .388 .009 .003 .400 x 1.0 mg/kg .485 .010 .005 .500 2.0 mg/kg .090 .008 .002 .100 .963 .027 .010 1.000 a) Are X and Y independent random variables? Explain. b) What is the probability that a randomly selected rat has: (i) one tumor, and (ii) at least one tumor? c) For a randomly selected rat in the 1.0 mg/kg drug dosage group, what is the probability that it has: (i) no tumor, (ii) at least one tumor? d) What is the expected number of tumors for a randomly selected rat in the 1.0 mg/kg drug dosage group? e) Does the 1.0 mg/kg drug dosage increase or decrease the expected number of tumors over the 0.0 mg/kg drug dosage? Justify your answer. f) You are given: E ( X ) = . 7, E ( X 2 ) = . 9, E ( Y ) = . 047, E ( Y 2 ) = . 067. Find: (i) Cov ( X,Y ), and (ii) X,Y . 4. Let X be defined by the probability density function f ( x ) =  2 x 1 < x 2 x < x 1 otherwise a) Find E ( X 3 ). b) Define Y = X 2 and find cov ( X,Y ). c) Are X and Y independent? Why or why not? 5. Let X and Y be defined by the joint probability density function given below: 2 f ( x,y ) = ( 2 e x y x y < otherwise a) Find P ( X + Y 3). b) Find the marginal pdfs of Y and X . b) Are X and Y independent? Justify your answer. 6. Let X take the value 0 if a child under 5 uses no seat belt, 1 if it uses adult seat belt, and 2 if it uses child seat. And let Y take the value 0 if a child survived a motor vehicle accident, and 1 if it did not. An extensive study undertaken by the National Highway Traffic Safety Administration resulted in the following conditional distributions of Y given X = x : y  1 P(Y=yx=0)  .69 .31 P(Y=yx=1)  .85 .15 P(Y=yx=2)  .84 .16 while the marginal distribution of X is x  1 2 P(X=x)  .54 .17 .29 (a) Use the table of conditional distributions of...
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This note was uploaded on 01/11/2012 for the course STAT 401 taught by Professor Akritas during the Fall '00 term at Pennsylvania State University, University Park.
 Fall '00
 Akritas

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