Question 1178446: Let X and Y be independent chi-square random variables with 14 and 5 degrees of freedom, respectively. Find
(a) P (|X − Y| ≤ 11.15),
(b) P (|X − Y| ≥ 3.8)
thank you
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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.
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