To find the mean and standard deviation of the given probability distribution, we use the following formulas:

1. Mean (μ): $\mu = \sum [x \cdot P(x)]$ Where $x$ is the value and $P(x)$ is the probability of that value.

2. Variance (σ²): $\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]$

3. Standard Deviation (σ): $\sigma = \sqrt{\sigma^2}$

Step 1: Calculate the mean (μ)

$\mu = (0 \times 0.05) + (4 \times 0.24) + (5 \times 0.32) + (7 \times 0.16) + (8 \times 0.23)$ I'll calculate that first.

Step 2: Calculate the variance (σ²) after finding the mean.

Let me calculate these.### Mean (μ): The mean of the given probability distribution is approximately 5.52.

Standard Deviation (σ):

The standard deviation of the given probability distribution is approximately 1.98.

Would you like more details on any of the steps? Here are five related questions to deepen your understanding:

1. How is the variance related to standard deviation?
2. What does the standard deviation tell us about a distribution?
3. How would the mean change if we altered one of the probabilities?
4. Can you compute the variance directly without finding the mean first?
5. How does a discrete probability distribution differ from a continuous one?

Tip: When the standard deviation is low, the data points are close to the mean, indicating less variability in the distribution.