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) = □

Solution Steps

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: $$E(2 + 4Z) = 2 + 4 \cdot 0 = 2$$

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: $$E\left(\frac{3Y - 4Z}{-5}\right) = \frac{3 \cdot 2 - 4 \cdot 0}{-5} = \frac{6}{-5} = -1.2$$

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: $$-5 + \text{Var}(2Y) = -5 + 188 = 183$$

Final Answer