STAT 400 Lecture AL1 Answers 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 a y 0 1 2 1 0 15 0 10 0 2 0 25 0 30 0 20 Find P X Y P X Y p 1 0 p 2 0 p 2 1 0 15 0 25 0 30 0 70 b Find P X Y 2 P X Y 2 p 1 1 p 2 0 0 10 0 25 0 35 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 y pY y x pX x 0 0 40 1 0 25 1 0 40 2 0 75 2 0 2 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 g x y f x y dx dy Find E X E Y E X Y E X Y E X 1 0 25 2 0 75 1 75 E Y 0 0 40 1 0 40 2 0 20 0 8 E X Y 1 0 15 2 0 25 2 0 10 3 0 30 3 0 4 0 20 2 55 OR E X Y E X E Y 1 75 0 8 2 55 E X Y 0 0 15 0 0 25 1 0 10 2 0 30 2 0 4 0 20 1 5 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 a 0 otherwise Verify that f x y is a legitimate probability density function f x y 0 for all x y 1 1 1 x 1 2 2 2 f x y dx dy 60 x y dy dx 30 x 1 x dx 0 0 0 2 30 x 2 60 x 3 30 x 4 dx 10 x 3 15 x 4 6 x 5 10 1 1 0 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 0 5 x 2 60 x y d y 0 0 0 5 P X Y 0 50 7 5 x 2 30 x 3 30 x 4 dx 2 5 x 3 7 5 x 4 6 x 5 00 5 321 0 03125 0 5 0 5 2 2 dx 30 x 0 5 x d x 0 0 c Find the probability that there are more almonds than cashews in a can That is find the probability P X Y 1 2 1 y 2 P X Y 60 x y dx dy 0 y 1 y 2 20 y 3 x dx dy y 0 12 20 y 1 y 12 3 y 3 dy 0 12 20 y 1 3 y 3 y 2 2 y 3 dx 20 y 60 y 12 2 60 y 3 40 y 4 dx 0 0 10 y 2 20 y 3 15 y 4 8 y 5 11 102 16 0 6875 OR 1 2 1 x 12 1 x 2 P X Y 1 60 x y dy dx 1 30 x 2 2 y dy dx 0 0 x x 12 1 0 12 1 30 x 2 1 x 2 x 2 dx 12 1 30 x 2 1 2 x dx 0 30 x 2 60 x 3 dx 1 10 x 3 15 x 4 0 11 102 16 OR 1 1 x 2 P X Y 60 x y dy dx 60 x 2 y dy dx 0 0 1 2 0 1 2 x 0 6875 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 1 P Y 2 X 1 3 1 x 60 x 2 y dy dx 0 2x 2 2 2 30 x 1 x 2 x dx 3 0 1 30 x 3 2 60 x 3 90 x 4 dx 0 1 3 30 x 2 60 x 3 90 x 4 dx 10 x 3 15 x 4 18 x 5 103 0 e 10 15 18 1 27 81 243 9 Find the marginal probability density function for X 1 x f X x 60 x 2 y dy 30 x 2 1 x 2 y dy 0 f 0 x 1 0 Find the marginal probability density function for Y 1 y 60 x f Y y 1 y 2 y dx 20 y 0 g 30 x 2 1 x 2 3x 2 dx 20 y 1 y 3 0 Find E X E Y E X Y E X Y 1 E X x 30 x 1 x 2 30 x 1 2 dx 3 60 x 4 30 x 5 dx 0 0 7 5 x 4 12 x 5 5 x 6 10 0 5 1 2 0 y 1 1 E Y y 20 y 1 y dy 3 20 y 1 2 60 y 3 60 y 4 20 y 5 dy 0 0 1 20 1 20 3 y 15 …
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