Question

1. Let X and Y be two discrete random variables each with the same the possible outcomes {1,2,3} a) Construct a bivariate probability mass function Px.y : {1,2,3} x {1,2,3} + R that satisfies the following properties propeties: (i) The expectation of X is E[X] = 2.1, (ii) The conditional expectation of Y given 2 = 3 is EY 2 = 3] = 1, (iii) The correlation between X and Y is slightly positive so that 0 < corr(X,Y) < 0.5 (Partial marks will be awarded for distributions satisfying only some of the above desiderata.) b) What is the correlation between X and Y for your proposed bivariate distribution? [15 marks)

Answer 1

Solution - Given that :- X and y core discrete random variables a) The Alabality distritution of Xandy we first need to define the support of x and y $={1,2,3,4} Sa= {1,2,344 Then let (x,y) become one at the possible outcomes of one tose of the Poeld of 'Choice continuity (1,1) outcome as (1, 2) (1 : 3) (114) P= (x=2, Y=y) = 16 The identitical probability for each canly) the probability - function The mean of eu calculated as KES yesz Ees. It (2,4) -I (1/16) + ---C116) +--+4) +---+4 El2)= { which one implity to EQ): 164/16) +26 4/16) +3 (4/16) +4C4/16) = -4/16=2.5. - 6) The mean of vis similarly calalated as core relation the Xandy = ECV) = Esi Eyes y+014) '(416)+ ---t\(116) +-- tu (1/16) + -- +4 which simplifies to ely) - 1 C4116) + 2 (4116) +364/16+424/16 =40/16 -2.5 Scanned with CamScanner