Question

1. Consider the following probability mass function for the discrete joint probability distribution for random variables X and Y where the possible values for X are 0, 1, 2, and 3; and the possible values for Y are 0, 1, 2, 3, and 4. y p(x, y) 0 1 2 3 4 0 0.10 0.05 0.15 0 0.01 1 0.09 0.15 0.10 0.02 0 x 2 0 0.03 0.07 0.11 0.06 3 0.01 0 0.01 0.02 0.02 a.) What is pX(x), the pmf of the marginal distribution for X? b.) What is pY(y), the pmf of the marginal distribution for Y? c.) If we know in some context that x = 0, what is the probability that y = 1? d.) If we know in some context that y = 0, what is the probability that x = 1? e.) State the complete conditional distribution pY|X(y|x)? f.) State the complete conditional distribution pX|Y(x|y)? g.) What are the two means of the marginal distributions ?x and ?Y? h.) What are the two standard deviations of the marginal distributions ?x and ?Y? i.) What is E(XY), the expected value of X times Y? j.) What is the covariance Cov(X, Y)? k.) What is the correlation coefficient ?XY? What does this tell us about the relationship between the two random variables? l.) Are X and Y independent random variables? How do you know?

          1. Consider the following probability mass function for the discrete joint probability distribution for random variables X and Y where the possible values for X are 0, 1, 2, and 3; and the possible values for Y are 0, 1, 2, 3, and 4.

        y
p(x, y) 0    1    2    3    4
    0   0.10 0.05 0.15 0    0.01
    1   0.09 0.15 0.10 0.02 0
x   2   0    0.03 0.07 0.11 0.06
    3   0.01 0    0.01 0.02 0.02

a.) What is pX(x), the pmf of the marginal distribution for X?
b.) What is pY(y), the pmf of the marginal distribution for Y?
c.) If we know in some context that x = 0, what is the probability that y = 1?
d.) If we know in some context that y = 0, what is the probability that x = 1?
e.) State the complete conditional distribution pY|X(y|x)?
f.) State the complete conditional distribution pX|Y(x|y)?
g.) What are the two means of the marginal distributions ?x and ?Y?
h.) What are the two standard deviations of the marginal distributions ?x and ?Y?
i.) What is E(XY), the expected value of X times Y?
j.) What is the covariance Cov(X, Y)?
k.) What is the correlation coefficient ?XY? What does this tell us about the relationship between the two random variables?
l.) Are X and Y independent random variables? How do you know?

1. Consider the following probability mass function for the discrete joint probability distribution for random variables X and Y where the possible values for X are 0, 1, 2, and 3; and the possible values for Y are 0, 1, 2, 3, and 4.

y
p(x, y) 0 1 2 3 4
0 0.10 0.05 0.15 0 0.01
1 0.09 0.15 0.10 0.02 0
x 2 0 0.03 0.07 0.11 0.06
3 0.01 0 0.01 0.02 0.02

a.) What is pX(x), the pmf of the marginal distribution for X?
b.) What is pY(y), the pmf of the marginal distribution for Y?
c.) If we know in some context that x = 0, what is the probability that y = 1?
d.) If we know in some context that y = 0, what is the probability that x = 1?
e.) State the complete conditional distribution pY|X(y|x)?
f.) State the complete conditional distribution pX|Y(x|y)?
g.) What are the two means of the marginal distributions ?x and ?Y?
h.) What are the two standard deviations of the marginal distributions ?x and ?Y?
i.) What is E(XY), the expected value of X times Y?
j.) What is the covariance Cov(X, Y)?
k.) What is the correlation coefficient ?XY? What does this tell us about the relationship between the two random variables?
l.) Are X and Y independent random variables? How do you know?

Added by John H.

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