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 ) = yx Ayxp,, . 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 ) = Adydxyxf ,. 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 ) = yyxpall,, p Y ( y ) = xyxpall,. The marginal probability density functions of X and of Y are given by f X ( x ) = , dyyxf, f Y ( y ) = , dxyxf. c) Find the (marginal) probability distributions p X ( x ) of X and p Y ( y ) of Y. y p Y ( y ) x p X ( 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 continuous E ( g( X, Y ) ) = x y yxpyxg all all),(),( E ( g( X, Y ) ) = dydxyxfyxg ),(),( d) Find E ( X ), E ( Y ), E ( X + Y ), E ( X Y ).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 otherwise01 ,10 ,10 60 ,2 yxyxyxyxf a) Verify that yxf , 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
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