Financial Risk Manager Part 1

The Central Limit Theorem (CLT) states that if $X_1, X_2, \ldots, X_n$ is a sequence of iid random variables with a finite mean $\mu$ and finite non-zero variance $\sigma^2$, then the distribution of $\frac{\hat{\mu} - \mu}{\sigma / \sqrt{n}}$ tends to a standard normal distribution as $n \to \infty$.

Put simply: $\frac{\hat{\mu} - \mu}{\sigma / \sqrt{n}} \to N(0, 1)$

Where $\hat{\mu} = \bar{X}$ = Sample Mean.

Analysis of options:

• Option A: Correct - This accurately describes the CLT with the proper denominator ($\sqrt{n}$) and limit as $n \to \infty$
• Option B: Incorrect - The limit is $n \to 0$ instead of $n \to \infty$
• Option C: Incorrect - The denominator is $\sigma\sqrt{n}$ instead of $\sqrt{n}$ and the limit is $n \to 1$ instead of $n \to \infty$
• Option D: Incorrect - Since option A is correct

Key points:

1. The CLT requires independent and identically distributed (iid) random variables
2. The variables must have finite mean ($\mu$) and finite non-zero variance ($\sigma^2$)
3. The standardized sample mean converges to a standard normal distribution as sample size approaches infinity
4. The standardization is: $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$ as $n \to \infty$