The probability density function plot for a Gaussian distribution has a symmetric bell shape, where $$\sigma$$ defines the width of the curve and $$\mu$$ describes the center of the curve. The y-axis represents the probability density. The curve shown below illustrates a standard normal distribution, which has a mean of $$\mu = 0$$ and a standard deviation of $$\sigma = 1$$.

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A normal distribution, also known as a Gaussian distribution, is a continuous probability distribution defined by its mean $$\mu$$ and variance $$\sigma^2$$ (where $$\sigma$$ is the standard deviation). The probability density function is given by the formula: $$p(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{1}{2 \sigma^2} (x - \mu)^2ight)$$. Changing the mean $$\mu$$ corresponds to a shift of the distribution along the $$x$$-axis, while increasing the variance $$\sigma^2$$ spreads the distribution out and lowers its peak.

Gaussian (Normal) Distribution

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

When we have $m$ $i.i.d$ samples  $\{x^{(1)},...,x^{(m)}\}$ from a Normal distribution with parameter $\mu$ and $\sigma$, such that: 
	$$P(x^{(i)};\mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$$,
a unbiased estimator for $\mu$ is the mean of those samples:
	$\hat{\mu}_m = \frac{1}{m} \sum_{i=1}^m x^{(i)}$


Unbiased Estimator for Normal $\mu$ Parameter

The mathematical formula for the normal distribution's probability density function can be directly implemented in Python. Using standard libraries like `math` and `numpy` (or their deep learning equivalents in PyTorch, JAX, or TensorFlow), a function can be defined to calculate the probability density $$p$$ for a given value $$x$$, given a mean $$\mu$$ and standard deviation $$\sigma$$. The code implementation is as follows:

```python
def normal(x, mu, sigma):
    p = 1 / math.sqrt(2 * math.pi * sigma**2)
    return p * np.exp(-0.5 * (x - mu)**2 / sigma**2)
```

This runnable implementation aligns with the formal definition: $$p(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{1}{2 \sigma^2} (x - \mu)^2\right)$$.

Normal Distribution Probability Density Function Code

The Central Limit Theorem is a foundational mathematical principle in probability theory stating that the average of $$n$$ independent random samples drawn from any distribution (with mean $$\mu$$ and standard deviation $$\sigma$$) will approximately follow a normal distribution centered at the true mean $$\mu$$ with a standard deviation of $$\sigma/\sqrt{n}$$, as the sample size $$n$$ grows large. In machine learning, this theorem explains why the empirical error of a classifier evaluated on a test set converges to its true population error at an asymptotic rate of $$\mathcal{O}(1/\sqrt{n})$$.

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