Consider the discrete random variable $X$ with the following distribution.

1. Explain why this is a valid probability model.

   Hint (a):

   In a valid probability model, the sum of the probabilities of the outcomes must be $1$. Also, the probability of each outcome must be between $0$ and $1$, inclusive.

2. Create a relative frequency histogram for this probability distribution.

3. Calculate the expected value of $X$.

   Step (c):

   $5(0.2) + 6(0.1) + 8(0.3) + 12(0.4)$