To calculate the discrete standard deviation for the given probability distribution, follow these steps:

Data Provided:

• $x$ values: 0, 1, 2, 3, 4
• $P(X)$ values (probabilities): 0.06, 0.10, 0.66, 0.08, 0.10

Formula for Standard Deviation (σ):

1. Mean (Expected Value): $\mu = E(X) = \sum (x \cdot P(X))$

2. Variance: $\sigma^2 = \sum \left( (x - \mu)^2 \cdot P(X) \right)$

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

Step 1: Calculate the Mean ($\mu$):

$\mu = (0 \cdot 0.06) + (1 \cdot 0.10) + (2 \cdot 0.66) + (3 \cdot 0.08) + (4 \cdot 0.10)$ $\mu = 0 + 0.10 + 1.32 + 0.24 + 0.40 = 2.06$

Step 2: Calculate the Variance ($\sigma^2$):

$\sigma^2 = \sum \left( (x - \mu)^2 \cdot P(X) \right)$ For each $x$:

• $(0 - 2.06)^2 \cdot 0.06 = 4.2436 \cdot 0.06 = 0.254616$
• $(1 - 2.06)^2 \cdot 0.10 = 1.1236 \cdot 0.10 = 0.11236$
• $(2 - 2.06)^2 \cdot 0.66 = 0.0036 \cdot 0.66 = 0.002376$
• $(3 - 2.06)^2 \cdot 0.08 = 0.8836 \cdot 0.08 = 0.070688$
• $(4 - 2.06)^2 \cdot 0.10 = 3.7636 \cdot 0.10 = 0.37636$

Now sum these: $\sigma^2 = 0.254616 + 0.11236 + 0.002376 + 0.070688 + 0.37636 = 0.8164$

Step 3: Calculate the Standard Deviation ($\sigma$):

$\sigma = \sqrt{0.8164} \approx 0.9035$

Thus, the discrete standard deviation is approximately 0.9035.

Would you like further details or clarification?

Here are five related questions for further exploration:

1. How does the mean (expected value) affect the standard deviation?
2. Can you calculate the variance without first calculating the mean?
3. How does the shape of the probability distribution affect the standard deviation?
4. What is the significance of standard deviation in probability theory?
5. How would the standard deviation change if the probabilities were adjusted?

Tip: When solving for standard deviation, always ensure the probabilities sum to 1, as they must represent a valid probability distribution.