Question

1. [12 points] ( X ) is a discrete uniform ( [-1,1] ) random variable, and ( Y=X^{2} ). The joint PMF of ( X ) and ( Y ) is given by [ P_{X, Y}(x, y)=left{egin{array}{cc} 1 / 3 & x=-1, y=1, \ 1 / 3 & x=0, y=0, \ 1 / 3 & x=1, y=1, \ 0 & ext { otherwise. } end{array} ight. ] a. Determine the marginal PMF ( P_{Y}(y) ). b. Determine ( operatorname{Cov}[X, Y] ). c. Are ( X ) and ( Y ) independent? Justify your answer.

          1. [12 points] ( X ) is a discrete uniform ( [-1,1] ) random variable, and ( Y=X^{2} ). The joint PMF of ( X ) and ( Y ) is given by
[
P_{X, Y}(x, y)=left{egin{array}{cc}
1 / 3 & x=-1, y=1, \
1 / 3 & x=0, y=0, \
1 / 3 & x=1, y=1, \
0 & 	ext { otherwise. }
end{array}
ight.
]
a. Determine the marginal PMF ( P_{Y}(y) ).
b. Determine ( operatorname{Cov}[X, Y] ).
c. Are ( X ) and ( Y ) independent? Justify your answer.

1. [12 points] ( X ) is a discrete uniform ( [-1,1] ) random variable, and ( Y=X^2 ). The joint PMF of ( X ) and ( Y ) is given by
[
PX, Y(x, y)=lefteginarraycc
1 / 3 x=-1, y=1, 1 / 3 x=0, y=0, 1 / 3 x=1, y=1, 0 ext otherwise.
endarray
ight.
]
a. Determine the marginal PMF ( PY(y) ).
b. Determine ( operatornameCov[X, Y] ).
c. Are ( X ) and ( Y ) independent? Justify your answer.

Added by King D.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Kirsty Gledhill

04/12/2024

Step 1

The marginal PMF of \(Y\), \(P_Y(y)\), is found by summing the joint PMF \(P_{X,Y}(x,y)\) over all possible values of \(X\). Given the joint PMF, we can see that \(Y\) can take on two values: 0 and 1. - For \(y=0\), the only pair \((x,y)\) that results in \(y=0\) Show more…

Show all steps