Question

# Let LaTeX: X,YX , Y be two discrete random variables that have the following joint distribution:...

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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