Fill up the table for me with the solution and answers

import pandas as pd

# Prepare the table columns
data = {
    'X': X,
    'P(X)': P_X,
    'X * P(X)': [x * p for x, p in zip(X, P_X)],
    '(X - μ)': [x - mean for x in X],
    '(X - μ)^2 * P(X)': [(x - mean) ** 2 * p for x, p in zip(X, P_X)]
}

# Create the DataFrame
df = pd.DataFrame(data)
dfHere is the filled table with the calculated values:

| X  | P(X)  | X * P(X) | (X - μ) | (X - μ)² * P(X) |
|----|-------|----------|---------|-----------------|
| 0  | 0.07  | 0.00     | -2.4    | 0.4032          |
| 1  | 0.18  | 0.18     | -1.4    | 0.3528          |
| 2  | 0.38  | 0.76     | -0.4    | 0.0608          |
| 3  | 0.12  | 0.36     | 0.6     | 0.0432          |
| 4  | 0.15  | 0.60     | 1.6     | 0.3840          |
| 5  | 0.10  | 0.50     | 2.6     | 0.6760          |

Total for \( X \cdot P(X) \) (Mean \( \mu \)): **2.4**  
Total for \( (X - \mu)^2 \cdot P(X) \) (Variance \( \sigma^2 \)): **1.92**

### Do you have any other questions?

Here are 5 additional questions you might explore:
1. How does altering the probabilities affect the distribution's mean and variance?
2. Why is variance a good measure of dispersion in a dataset?
3. Can two different distributions have the same mean but different variances?
4. What happens to the standard deviation when all probabilities are the same?
5. How can we visualize this probability distribution using a graph?

**Tip:** It's useful to create visual representations (such as histograms) to better understand how data is distributed.

Explore a detailed analysis of a probability distribution, focusing on calculating mean (expected value) and variance. Understand how altering probabilities affects distribution characteristics. Includes a table with computed values and relevant mathematical concepts.

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