Let LaTeX: X,YX , Y be two discrete random variables that have the following joint distribution:
x = 0 1
y = -1 0.18 0.12
0 ? 0.20
1 0.12 0.08
(a) Determine the following probabilities:
LaTeX: P(X=0, Y=0) P ( X = 0 , Y = 0 ), LaTeX: P(X\le 0,Y\le 0)P ( X ≤ 0 , Y ≤ 0 )
(b) Find the marginal distribution of LaTeX: YY.
(c) What is the conditional distribution of LaTeX: XX given LaTeX: Y=-1Y = − 1?
(d) Are LaTeX: X,YX , Y independent? State your reason clearly.
(e) What is LaTeX: E\left(XY\right)E ( X Y )?
The joint distribution is given to be as follows,
(with X on the horizontal and Y on the vertical). Here the missing value is obtained by subtracting the sum of all other values from 1 since the total probability must be equal to 1. Then,
a)
b) The marginal distribution of Y is obtained as,
c) Conditional distribution of X given Y = -1,
d) Expected value of XY.
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