To find the average of a discrete probability distribution, follow these simple steps:

1. List the Outcomes: Write down all the possible results, like $x_1, x_2, ..., x_n$.

2. Find Their Probabilities: For each outcome, write down its chance of happening, like $P(x_1), P(x_2), ..., P(x_n)$.

3. Use the Mean Formula: To get the average, use this formula:

$$\mu = \sum (x_i \cdot P(x_i))$$

Here, $\mu$ means the average.

Example: Let's say we have these outcomes and their probabilities:

• Outcome $1$: $P(1) = 0.2$
• Outcome $2$: $P(2) = 0.5$
• Outcome $3$: $P(3) = 0.3$

To find the average:

$$\mu = (1 \cdot 0.2) + (2 \cdot 0.5) + (3 \cdot 0.3) = 0.2 + 1 + 0.9 = 2.1$$

So, the average of this distribution is $2.1$.