STAT 462 Homework 7 due Thursday, November 1, 2012 1. A pdf is defined by ( 2 ) if 0 <1><> Y ) if X and Y are jointly distributed with pdf , ( , )= , 0 1, 0 1. X Y f x y x ? y ? x ? ? y ? (b) Find P(X 2 <><>
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STAT 462 Homework 7 due Thursday, November 1, 2012 1. A pdf is defined by Cx (?2y) if 0<> Y) if X and Y are jointly distributed with pdf fx (,y)=x?y,0??x 1,0?y?1. XY , 2 (b) Find PX(<1,>
Microsoft Word - homework7 STAT 462 Homework 7 due Thursday, November 1, 2012 1. A pdf is defined by ( 2 ) if 0 <>< 1="" and="" 0=""><>< 2="" (="" ,="" )="0" otherwise="" c="" x="" y="" y="" x="" f="" x="" y="" ="" ="" ="" (a)="" find="" the="" value="" of="" c="" .="" (b)="" find="" the="" marginal="" distribution="" of="" x="" .="" (c)="" find="" the="" joint="" cdf="" of="" x="" and="" y="" .="" define="" for="" all="" .="" ="" 2.="" (a)="" find="" (=""> )P X Y if X and Y are jointly distributed with pdf , ( , ) = , 0 1, 0 1.X Yf x y x y x y (b) Find 2( <>< )p="" x="" y="" x="" if="" x="" and="" y="" are="" jointly="" distributed="" with="" pdf="" ,="" (="" ,="" )="2" ,="" 0="" 1,="" 0="" 1.x="" yf="" x="" y="" x="" x="" y="" ="" ="" ="" 3.="" define="" 2="" 2="" ,="" 21="" 0=""><><>< 1, 0 ( , ) = 2 0 x y x y x y x f x y otherwise find the marginal distributions of x and y . 4. let 1y and 2y denote the proportion of time (out of one workday) during which employees i and ii, respectively, perform their assigned tasks. the joint relative frequency behavior of 1y and 2y is modeled by the pdf 1 2 1 2 1 2 , 1 2 0 1, 0 1 ( , ) = . 0y y y y y y f y y otherwise (a) find 1 21 1, .2 4p y y (b) find 1 2 1 .p y y (c) find 1 2( 1/ 2 | 1/ 2)p y y . (d) find the marginal probability density functions for 1y and 2.y (e) are 1y and 2y independent? why or why not? (f) employee i has a higher productivity rating than employee ii and a measure of the total productivity of the pair of employees is 30 1y + 25 2y . find the expected value of this measure of productivity. (g) find the variance for the measure of productivity in (f). 5. the random variables x and y have the joint distribution xyf given by 1 1 1 0,1,.... , 0 ,1 0, 0! , . 0 x y xy y x x e y f x y else (a) calculate the marginal pdf ( ).xf x identify this distribution and its parameter(s). (b) calculate the marginal pmf ( ).yf y • book problems: 5.15, 5.52, 5.59 1,="" 0="" (="" ,="" )="2" 0="" x="" y="" x="" y="" x="" y="" x="" f="" x="" y="" otherwise="" ="" ="" ="" ="" find="" the="" marginal="" distributions="" of="" x="" and="" y="" .="" 4.="" let="" 1y="" and="" 2y="" denote="" the="" proportion="" of="" time="" (out="" of="" one="" workday)="" during="" which="" employees="" i="" and="" ii,="" respectively,="" perform="" their="" assigned="" tasks.="" the="" joint="" relative="" frequency="" behavior="" of="" 1y="" and="" 2y="" is="" modeled="" by="" the="" pdf="" 1="" 2="" 1="" 2="" 1="" 2="" ,="" 1="" 2="" 0="" 1,="" 0="" 1="" (="" ,="" )="." 0y="" y="" y="" y="" y="" y="" f="" y="" y="" otherwise="" ="" ="" ="" ="" ="" ="" ="" (a)="" find="" ="" 1="" 21="" 1,="" .2="" 4p="" y="" y="" ="" (b)="" find="" ="" 1="" 2="" 1="" .p="" y="" y="" ="" (c)="" find="" 1="" 2(="" 1/="" 2="" |="" 1/="" 2)p="" y="" y="" ="" .="" (d)="" find="" the="" marginal="" probability="" density="" functions="" for="" 1y="" and="" 2.y="" (e)="" are="" 1y="" and="" 2y="" independent?="" why="" or="" why="" not?="" (f)="" employee="" i="" has="" a="" higher="" productivity="" rating="" than="" employee="" ii="" and="" a="" measure="" of="" the="" total="" productivity="" of="" the="" pair="" of="" employees="" is="" 30="" 1y="" +="" 25="" 2y="" .="" find="" the="" expected="" value="" of="" this="" measure="" of="" productivity.="" (g)="" find="" the="" variance="" for="" the="" measure="" of="" productivity="" in="" (f).="" 5.="" the="" random="" variables="" x="" and="" y="" have="" the="" joint="" distribution="" xyf="" given="" by="" ="" ="" ="" ="" 1="" 1="" 1="" 0,1,....="" ,="" 0="" ,1="" 0,="" 0!="" ,="" .="" 0="" x="" y="" xy="" y="" x="" x="" e="" y="" f="" x="" y="" else="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" (a)="" calculate="" the="" marginal="" pdf="" (="" ).xf="" x="" identify="" this="" distribution="" and="" its="" parameter(s).="" (b)="" calculate="" the="" marginal="" pmf="" (="" ).yf="" y="" •="" book="" problems:="" 5.15,="" 5.52,="">