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Statistics 270 - Lecture 18 • Will begin Chapter 5 today • Many situations where one is interested in more than one random variable • Have a joint distribution for such cases Example • 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 Definition • 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 Discrete RV’s • Usual properties of pmf’s still hold Example • 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 Example • 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 Definition • 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 ) p( x, y ) y x Example • 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 Definition • 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 Continuous rv’s • Usual properties of pdf’s still hold Example: • 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 30 and 20 y 30 • Notice that they seem to vary jointly