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St a t i s t i c s 2 7 0 - L e c t u r e 1 8 • Will begin Chapter 5 today • Many situations where one is interested in more than one random variable • Have a joint distribution for such cases Ex a m p l e • Let X and Y be random variables with pmf x y 1 2 1 2 3 0.10 0.5 0.10 0.05 0.10 0.15 De f i n i t i o n • Let X and Y be rv’s on a sample space S • Discrete rv’s: The joint prob. mass function for each (x,y) is defined by p(x,y)= P(X= x, Y= y) • If A is an event then, P ( X , Y ) A p ( x, y ) ( x, y ) A Di s c r e t e RV ’s • Usual properties of pmf’s still hold Ex a m p l e • Let X and Y be random variables with pmf x y • Observations: • P(X= 2,Y= 2)= • P(X> 1, Y= 1) 1 2 1 2 3 0.10 0.5 0.10 0.05 0.10 0.15 Ex a m p l e • Let X be the number of Canon digital cameras sold in a week at a certain store • The pmf for X is x p(x) 0 .1 1 .2 2 .3 3 .25 4 .15 • 60% of all customers who purchase camera also purchase the longterm warranty • Determine the joint pdf of X and Y De f i n i t i o n • The marginal probability mass function for discrete random varaibles X and Y, denote by pX(x) and pY(y), respectively, are given by p X ( x) p ( x, y ) and pY ( y ) y p ( x, y ) x Ex a m p l e • Let X be the number of Canon digital cameras sold in a week at a certain store • The pmf for X is x p(x) 0 .1 1 .2 2 .3 3 .25 4 .15 • 60% of all customers who purchase camera also purchase the longterm warranty • Find the marginal distributions of X and Y De f i n i t i o n • Let X and Y be rv’s on a sample space S • Continuous rv’s: The joint prob. Distribution function for (x,y) is defined by f(x,y) • If A is an event then, P ( X , Y ) A f ( x, y )dxdy A Co n t i n u o u s r v ’s • Usual properties of pdf’s still hold Ex a m p l e : • The front tire on a particular type of car is suppose to be filled to a pressure of 26 psi • Suppose the actual air pressure in EACH tire is a random variable (X for the right side; Y for the left side) with joint pdf f ( x, y ) • K (x 2 y 2 ) for 20 x Notice that they seem to vary jointly 30 and 20 y 30 This document was created with Win2PDF available at http://www.daneprairie.com. The unregistered version of Win2PDF is for evaluation or non-commercial use only.