Signs and their sum

Topic: Joint distributions · more questions on this topic
‹ A point in a triangle

Let U and V be independent, with \mathbb{P}(U=1)=\mathbb{P}(V=1)=1/4 and \mathbb{P}(U=-1)=\mathbb{P}(V=-1)=3/4. Set X=U/V and Y=U+V.

  1. Give the joint probability mass function of (X,Y).

  2. Calculate \operatorname{Cov}(X,Y).

  1. Since V=\pm1, 1/V=V and X=UV. The four sign pairs give
(U,V) Probability X Y
(1,1) 1/16 1 2
(1,-1) 3/16 -1 0
(-1,1) 3/16 -1 0
(-1,-1) 9/16 1 -2

so \mathbb{P}(X=1,Y=2)=1/16, \mathbb{P}(X=-1,Y=0)=6/16 and \mathbb{P}(X=1,Y=-2)=9/16.

  1. \mathbb{E} X=10/16-6/16=1/4, \mathbb{E} Y=\mathbb{E} U+\mathbb{E} V=2(1/4-3/4)=-1 and \mathbb{E}(XY)=2\times1/16-2\times9/16=-1. So

\operatorname{Cov}(X,Y)=-1-\tfrac14\times(-1)=-\tfrac34.