Let the following be a joint probability mass function for the random variables

Question

Let the following be a joint probability mass function for the random variables X and Y.

a)Determine the marginal probability distribution of the random variables X and Y

b)Determine P(X1)

c) Determine P(Y<1.5)

d)   Are the random variables X and Y independent? Why or why not?

e)Determine the conditional probability distribution of Y given that X= 1

f)Calculate the correlation coefficient between X and Y

x

y

fxy(x,y)

0

1

1/8

1

0

1/8

1

1

1/4

2

2

1/2

x

y

fxy(x,y)

0

1

1/8

1

0

1/8

1

1

1/4

2

2

1/2

Explanation / Answer

b) P(X<=1) = 1/2

c) P(Y<1.5) = 1/2

e) P(Y/x=1)

is y        0     1     2

   p         1/8 1/4 0

--------------------------------------

f) E(XY) = sum of xyP(XY)

= 1/4 + 4(1/2)

= 9/4

E(X)E(Y) = (3/2)(3/2) = 9/4

Hence cov (xy) =0

Corre = 0

x y fxy(x,y) 0 1 1/8 1 0 1/8 1 1 1/4 2 2 1/2 PDF OF x x 0 1    2 Total prob 1/8 3/8 1/2 1    PMF of y y 0 1 2 Total prob 1/8 3/8 1/2 1    f(0,1) 1/8 f(0)f(1) 1/64 As the above two are not equal, x,y not independent.

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