Let's solve this step by step.

Step 1: Compute $xP(x)$ for each value of $x$.

Step 2: Find the mean $\mu$.

The mean is calculated as: $\mu = \sum xP(x)$ Adding up the $xP(x)$ values: $\mu = 0.00 + 0.20 + 0.42 + 0.36 + 1.24 = 2.22$

So, the mean $\mu$ is: $\mu = 2.22$

Step 3: Compute the values in the standard deviation table.

The formula for the standard deviation $\sigma$ is: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)}$ First, we compute each row step by step:

Step 4: Sum $(x - \mu)^2 \cdot P(x)$.

Add up all the values in the $(x - \mu)^2 \cdot P(x)$ column: $\text{Sum} = 0.7888 + 0.2980 + 0.0105 + 0.0732 + 0.9827 = 2.152$

Step 5: Find the standard deviation $\sigma$.

The standard deviation is: $\sigma = \sqrt{\text{Sum}}$ $\sigma = \sqrt{2.152} \approx 1.467$

Final Answers:

• Mean $\mu$: 2.22
• Standard Deviation $\sigma$: 1.467

Let me know if you need clarification or a deeper explanation!

Related Questions:

1. What is the variance, and how is it related to the standard deviation?
2. How would you interpret this mean and standard deviation in a real-world context?
3. What happens to $\mu$ and $\sigma$ if all probabilities $P(x)$ are equal?
4. How do we test if this probability distribution is valid?
5. What are the key differences between a mean and median in a probability distribution?

Tip:

To confirm your calculations, always ensure that the sum of $P(x)$ equals 1 for a valid probability distribution!