Question

2) Consider two random variables X and Y with a joint probability density function fXY(x, y) = {axy 0 < x < 1 and 0 < y < 1, 0 else} a) Determine the value of ?. b) Determine the marginal probability density function fX(x) of X. c) Determine the marginal probability density function fY(y) of Y. d) Determine whether the random variable X and Y are independent. 3) Consider two random variables X and Y with a joint probability mass function as shown in the table below: y -1 0 1 2 fXY(x, y) x -1 0.04 0.08 0.12 0.16 1 0.06 0.12 ? 0.24 Note that fXY(x, y) = 0 for all (x,y) not shown in the table. a) Determine the value of ?. b) Determine the marginal probability density function fX(x) of X. c) Determine the marginal probability density function fY(y) of Y. d) Determine whether the random variable X and Y are independent. 4) Consider the experiment of rolling two dice. Let the random variable X denote the maximum value of the two values on the two dice and Y the minimum value of the two values on the two dice. a) Determine the joint probability mass function fXY(x, y). b) Determine the marginal probability density function fX(x) of X. c) Determine the marginal probability density function fY(y) of Y. d) Determine whether the random variable X and Y are independent.

          2) Consider two random variables X and Y with a joint probability density function
fXY(x, y) = {axy 0 < x < 1 and 0 < y < 1, 0 else}
a) Determine the value of ?.
b) Determine the marginal probability density function fX(x) of X.
c) Determine the marginal probability density function fY(y) of Y.
d) Determine whether the random variable X and Y are independent.

3) Consider two random variables X and Y with a joint probability mass function as shown in the table below:

y
-1 0 1 2
fXY(x, y)
x
-1 0.04 0.08 0.12 0.16
1 0.06 0.12 ? 0.24

Note that fXY(x, y) = 0 for all (x,y) not shown in the table.
a) Determine the value of ?.
b) Determine the marginal probability density function fX(x) of X.
c) Determine the marginal probability density function fY(y) of Y.
d) Determine whether the random variable X and Y are independent.

4) Consider the experiment of rolling two dice. Let the random variable X denote the maximum value of the two values on the two dice and Y the minimum value of the two values on the two dice.
a) Determine the joint probability mass function fXY(x, y).
b) Determine the marginal probability density function fX(x) of X.
c) Determine the marginal probability density function fY(y) of Y.
d) Determine whether the random variable X and Y are independent.

2) Consider two random variables X and Y with a joint probability density function
fXY(x, y) = axy 0 < x < 1 and 0 < y < 1, 0 else
a) Determine the value of ?.
b) Determine the marginal probability density function fX(x) of X.
c) Determine the marginal probability density function fY(y) of Y.
d) Determine whether the random variable X and Y are independent.

3) Consider two random variables X and Y with a joint probability mass function as shown in the table below:

y
-1 0 1 2
fXY(x, y)
x
-1 0.04 0.08 0.12 0.16
1 0.06 0.12 ? 0.24

Note that fXY(x, y) = 0 for all (x,y) not shown in the table.
a) Determine the value of ?.
b) Determine the marginal probability density function fX(x) of X.
c) Determine the marginal probability density function fY(y) of Y.
d) Determine whether the random variable X and Y are independent.

4) Consider the experiment of rolling two dice. Let the random variable X denote the maximum value of the two values on the two dice and Y the minimum value of the two values on the two dice.
a) Determine the joint probability mass function fXY(x, y).
b) Determine the marginal probability density function fX(x) of X.
c) Determine the marginal probability density function fY(y) of Y.
d) Determine whether the random variable X and Y are independent.

Added by Christopher G.

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