To calculate the mean of a probability distribution, we use the formula:

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

The table in your image provides the following data:

• $x$: 0, 1, 2, 3
• $P(x)$: 0.25, 0.3, 0.25, 0.2

Step-by-Step Calculation:

1. Multiply each $x$ by its corresponding $P(x)$:

   • For $x = 0$: $0 \cdot 0.25 = 0$
   • For $x = 1$: $1 \cdot 0.3 = 0.3$
   • For $x = 2$: $2 \cdot 0.25 = 0.5$
   • For $x = 3$: $3 \cdot 0.2 = 0.6$

2. Add the results: $0 + 0.3 + 0.5 + 0.6 = 1.4$

Final Answer:

The mean of this probability distribution is: $\boxed{1.4}$

Let me know if you want further clarification or have any questions.

5 Related Questions:

1. How do you calculate the variance of a probability distribution?
2. What is the significance of the mean in probability and statistics?
3. How would the mean change if $P(x)$ values were adjusted?
4. What are the conditions for a valid probability distribution?
5. How can we interpret the mean of a discrete random variable?

Tip: Always verify that the probabilities $P(x)$ add up to 1 when working with probability distributions.