Central Limit Theorem: Simulation-Based Exploration

Theoretical Foundation

The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the sampling distribution of the mean of any independent random variable will be approximately normally distributed, regardless of the variable's original distribution.

Key Formula:

If $X_1, X_2, \dots, X_n$ are independent and identically distributed (i.i.d.) random variables with mean $\mu$ and variance $\sigma^2$, then the sample mean:

\[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \]

has an approximate normal distribution:

\[ \bar{X}_n \sim \mathcal{N}\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{as } n \to \infty \]

Analysis of Range

We explore the following distributions:

1. Uniform Distribution $\sim \mathcal{U}(a, b)$

  • Mean: \(\mu = \frac{a + b}{2}\)
  • Variance: \(\sigma^2 = \frac{(b - a)^2}{12}\)

2. Exponential Distribution $\sim \text{Exp}(\lambda)$

  • Mean: \(\mu = \frac{1}{\lambda}\)
  • Variance: \(\sigma^2 = \frac{1}{\lambda^2}\)

3. Binomial Distribution $\sim \text{Bin}(n, p)$

  • Mean: \(\mu = np\)
  • Variance: \(\sigma^2 = np(1 - p)\)

As $n$ increases, the sample mean distribution approaches:

\[ \bar{X}_n \approx \mathcal{N}(\mu, \frac{\sigma^2}{n}) \]

Practical Applications

  • Statistical inference: Confidence intervals and hypothesis tests rely on normal approximations.
  • Quality control: Monitoring product variation via sample averages.
  • Economics & finance: Portfolio returns modeled via CLT for large $n$.

Implementation with Python Simulation

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1. Simulating Sampling Distributions

🔢 Objective: Simulate three types of populations to explore how their shapes influence the behavior of sample means in the Central Limit Theorem (CLT).

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Python Code: CLT in Quality Control

alt text Diagram Explanation:

  • The histogram shows the distribution of the means of 1000 random samples, each with 30 product weights.

  • The red dashed line represents the average of these sample means (≈ population mean).

  • The bell-shaped curve demonstrates that sample means approximate a normal distribution, validating the Central Limit Theorem.

  • Explanation: With increasing samples, discrete distributions yield smooth, approximately normal sample mean distributions.

Conclusion

  • The CLT holds for all tested distributions regardless of their shape.
  • As sample size $n$ increases, the sample mean distribution converges to a normal distribution with mean $\mu$ and variance $\sigma^2/n$.
  • Practical experiments confirm theoretical expectations, reinforcing the foundational importance of the CLT in statistical analysis.