Suppose a random variable X has the following discrete probability mass function X = 0 with probability 1/3, 1 with probability 1/8 and 2 with probability 1/2.

The expected value of a discrete random variable is $E[X] = \sum x \cdot P(X=x)$.

$$E[X] = (0 \cdot P(X=0)) + (1 \cdot P(X=1)) + (2 \cdot P(X=2))$$ $$E[X] = \left(0 \cdot \frac{1}{3}\right) + \left(1 \cdot \frac{1}{8}\right) + \left(2 \cdot \frac{1}{2}\right)$$ $$E[X] = 0 + \frac{1}{8} + 1$$ $$E[X] = \frac{1}{8} + \frac{8}{8}$$ $$E[X] = \frac{9}{8}$$

The expected value of $X^2$ is $E[X^2] = \sum x^2 \cdot P(X=x)$.

$$E[X^2] = (0^2 \cdot P(X=0)) + (1^2 \cdot P(X=1)) + (2^2 \cdot P(X=2))$$ $$E[X^2] = \left(0 \cdot \frac{1}{3}\right) + \left(1 \cdot \frac{1}{8}\right) + \left(4 \cdot \frac{1}{2}\right)$$ $$E[X^2] = 0 + \frac{1}{8} + 2$$ $$E[X^2] = \frac{1}{8} + \frac{16}{8}$$ $$E[X^2] = \frac{17}{8}$$

The variance is given by $Var[X] = E[X^2] - (E[X])^2$.

$$Var[X] = \frac{17}{8} - \left(\frac{9}{8}\right)^2$$ $$Var[X] = \frac{17}{8} - \frac{81}{64}$$

To subtract, find a common denominator, which is 64.

$$Var[X] = \frac{17 \cdot 8}{8 \cdot 8} - \frac{81}{64}$$ $$Var[X] = \frac{136}{64} - \frac{81}{64}$$ $$Var[X] = \frac{136 - 81}{64}$$ $$Var[X] = \frac{55}{64}$$

For part I:
$E[X] = \boxed{\frac{9}{8}}$
$Var[X] = \boxed{\frac{55}{64}}$

II) Find $Var[5X - 2]$

Here, $a=5$ and $b=-2$.

$$Var[5X - 2] = 5^2 \cdot Var[X]$$ $$Var[5X - 2] = 25 \cdot \frac{55}{64}$$ $$Var[5X - 2] = \frac{25 \cdot 55}{64}$$ $$Var[5X - 2] = \frac{1375}{64}$$

For part II:
$Var[5X - 2] = \boxed{\frac{1375}{64}}$

III) Find $Var[7X + 47]$

Here, $a=7$ and $b=47$.

$$Var[7X + 47] = 7^2 \cdot Var[X]$$ $$Var[7X + 47] = 49 \cdot \frac{55}{64}$$ $$Var[7X + 47] = \frac{49 \cdot 55}{64}$$ $$Var[7X + 47] = \frac{2695}{64}$$

For part III:
$Var[7X + 47] = \boxed{\frac{2695}{64}}$