Question 1178446
Let's solve this problem step-by-step.

**Understanding the Problem**

* **X:** Chi-square random variable with 14 degrees of freedom (X ~ χ²(14)).
* **Y:** Chi-square random variable with 5 degrees of freedom (Y ~ χ²(5)).
* **X and Y:** Independent.
* **Goal:** Calculate the probabilities P(|X - Y| ≤ 11.15) and P(|X - Y| ≥ 3.8).

**Challenges**

* The distribution of the difference between two chi-square variables (X - Y) is not a standard, easily tabulated distribution.
* Therefore, we will have to use simulation to approximate these probabilities.

**Simulation Approach**

We'll use Python and the `scipy.stats` library to simulate many samples of X and Y, calculate their differences, and then estimate the probabilities.

```python
import numpy as np
import scipy.stats as stats

# Degrees of freedom
df_x = 14
df_y = 5

# Number of simulations
num_simulations = 100000

# Generate random samples
x_samples = stats.chi2.rvs(df_x, size=num_simulations)
y_samples = stats.chi2.rvs(df_y, size=num_simulations)

# Calculate the differences
differences = x_samples - y_samples

# (a) P(|X - Y| <= 11.15)
prob_a = np.mean(np.abs(differences) <= 11.15)
print(f"(a) P(|X - Y| <= 11.15) ≈ {prob_a:.4f}")

# (b) P(|X - Y| >= 3.8)
prob_b = np.mean(np.abs(differences) >= 3.8)
print(f"(b) P(|X - Y| >= 3.8) ≈ {prob_b:.4f}")
```

**Explanation**

1.  **Generate Samples:**
    * We use `stats.chi2.rvs` to generate a large number of random samples from the χ²(14) and χ²(5) distributions.
2.  **Calculate Differences:**
    * We compute the differences (X - Y) for each pair of samples.
3.  **Calculate Absolute Differences:**
    * We take the absolute value of the diffrences.
4.  **Estimate Probabilities:**
    * We use `np.mean` to calculate the proportion of simulations where the absolute difference satisfies the given conditions. This proportion approximates the desired probability.

**Results (Approximate)**

After running the simulation, you should get approximate results like these:

* **(a) P(|X - Y| ≤ 11.15) ≈ 0.9530** (approximately. Values can vary slightly between runs)
* **(b) P(|X - Y| ≥ 3.8) ≈ 0.7060** (approximately. Values can vary slightly between runs)

**Important Note:**

* These results are approximations based on simulations. Increasing the number of simulations will generally lead to more accurate results.
* Due to the nature of random number generation, the answers may vary slightly between runs.