Questions: Random Variables and Distributions Suppose that x, y, and z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following: E(X) = 6 E(Y) = 2 E(Z) = 0 Var(X) = 36 Var(Y) = 47 Var(Z) = 7 Compute the values of the expressions below. E(2 + 4Z) = □ E((3Y - 4Z)/-5) = □ -5 + Var(2Y) = □ E(4Y^2) = □

Random Variables and Distributions

Suppose that x, y, and z are jointly distributed random variables, that is, they are defined on the same sample space.

Suppose that we also have the following:
E(X) = 6 E(Y) = 2 E(Z) = 0
Var(X) = 36 Var(Y) = 47 Var(Z) = 7

Compute the values of the expressions below.

E(2 + 4Z) = □

E((3Y - 4Z)/-5) = □

-5 + Var(2Y) = □

E(4Y^2) = □

Transcript text: Random Variables and Distributions Suppose that x, y, and z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following: E(X) = 6 E(Y) = 2 E(Z) = 0 Var(X) = 36 Var(Y) = 47 Var(Z) = 7 Compute the values of the expressions below. E(2 + 4Z) = □ E(\frac{3Y - 4Z}{-5}) = □ -5 + Var(2Y) = □ E(4Y^2) = □

Solution

Solution Steps

Step 1: Compute \( E(2 + 4Z) \)

The expectation of a linear combination of random variables is given by: \[ E(a + bZ) = a + bE(Z) \] Here, \( a = 2 \) and \( b = 4 \). Given \( E(Z) = 0 \), we have: \[ E(2 + 4Z) = 2 + 4 \cdot 0 = 2 \]

Step 2: Compute \( E\left(\frac{3Y - 4Z}{-5}\right) \)

The expectation of a linear combination of random variables is given by: \[ E\left(\frac{aY + bZ}{c}\right) = \frac{aE(Y) + bE(Z)}{c} \] Here, \( a = 3 \), \( b = -4 \), and \( c = -5 \). Given \( E(Y) = 2 \) and \( E(Z) = 0 \), we have: \[ E\left(\frac{3Y - 4Z}{-5}\right) = \frac{3 \cdot 2 - 4 \cdot 0}{-5} = \frac{6}{-5} = -1.2 \]

Step 3: Compute \( -5 + \text{Var}(2Y) \)

The variance of a scaled random variable is given by: \[ \text{Var}(aY) = a^2 \text{Var}(Y) \] Here, \( a = 2 \). Given \( \text{Var}(Y) = 47 \), we have: \[ \text{Var}(2Y) = 2^2 \cdot 47 = 4 \cdot 47 = 188 \] Thus: \[ -5 + \text{Var}(2Y) = -5 + 188 = 183 \]