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

a) To find pdf of x and y

Marginal density of x x 0 1 2 Total p 1/8 3/8 1/2 1      p*x 0      3/8 1      1 3/8 p*x^2 0      3/8 2      2 3/8 Variance 31/64 Marginal density of y y 0 1 2 Total p 1/8 3/8 1/2 1      p*y 0      3/8 1      1 3/8 b) P(X<=1) 1/2 c) P(Y<1.5) 1/2 d) P(0,1) = 1/8     P(X=0)P(Y=1) 3/64 Since the two are not the same x,y are not independent. e) P(Y/x=1) y 0 1 2 P(Y/x=1) 1/8 1/4 0     

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