STAT 400 Lecture AL1 Examples for 4 1 4 4 Part 1 Spring 2015 Dalpiaz Multivariate Distributions Let X and Y be two discrete random variables The joint probability mass function p x y is defined for each pair of numbers x y by p x y P X x and Y y Let A be any set consisting of pairs of x y values Then P X Y A p x y x y A Let X and Y be two continuous random variables Then f x y is the joint probability density function for X and Y if for any two dimensional set A P X Y A f x y dx dy A 1 Consider the following joint probability distribution p x y of two random variables X and Y x y 0 1 2 1 0 15 0 10 0 2 0 25 0 30 0 20 a Find P X Y 2 b Find P X Y The marginal probability mass functions of X and of Y are given by p X x p x y p Y y p x y all x all y The marginal probability density functions of X and of Y are given by f Y y f x y dx f X x f x y dy c Find the marginal probability distributions p X x of X and p Y y of Y x y pY y pX x If p x y is the joint probability mass function of X Y OR f x y is the joint probability density function of X Y then discrete E g X Y g x y p x y continuous E g X Y all x all y d Find E X E Y E X Y E X Y g x y f x y dx dy 2 Alex is Nuts Inc markets cans of deluxe mixed nuts containing almonds cashews and peanuts Suppose the net weight of each can is exactly 1 lb but the weight contribution of each type of nut is random Because the three weights sum to 1 a joint probability model for any two gives all necessary information about the weight of the third type Let X the weight of almonds in a selected can and Y the weight of cashews Then the region of positive density is D x y 0 x 1 0 y 1 x y 1 Let the joint probability density function for X Y be 2 f x y 60 x y 0 x 1 0 y 1 x y 1 0 otherwise a Verify that f x y is a legitimate probability density function b Find the probability that the two types of nuts together make up less than 50 of the can That is find the probability P X Y 0 50 Find the probability that peanuts make up over 50 of the can c Find the probability that there are more almonds than cashews in a can That is find the probability P X Y d Find the probability that there are at least twice as many cashews as there are almonds That is find the probability P 2 X Y e Find the marginal probability density function for X f Find the marginal probability density function for Y g Find E X E Y E X Y E X Y h If 1 lb of almonds costs the company 1 00 1 lb of cashews costs 1 50 and 1 lb of peanuts costs 0 60 what is the expected total cost of the content of a can
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