Empirical distribution  showcases the empirical frequency associated with a point/sample in the dataset/set of samples.

$\hat{p}(x) = \frac{1}{m} \sum_{i=1}^{m} \delta (x-x^i)$

The dirac delta function is used to describe the empirical distribution of continuous random variables.

Multinoulli distribution is sufficient to explain the empirical distribution of discrete random variables.


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Common probability distributions include Normal Distribution, Multinoulli Distribution,  Bernoulli Probability Distribution, Exponential Distribution, and Laplace Distribution. 

Common Probability Distributions

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

Deep Learning

Bernoulli distribution is a distribution over a single binary variable. It can only have results in 0 or 1. The sum of the probability of 0  and the probability of 1 is equal to 1. If we denote the probability of 0 as $\sigma$, Bernoulli distribution then has a mean of $\sigma$ and a variance of  $\sigma * (1- \sigma)$.

Bernoulli Distribution

Laplace distribution also called double exponential distribution has a sharp bend at random values of  $\mu$

$$Laplace(x;\mu,\gamma)= \frac{1}{2 \gamma} e^{\frac{ - \lvert x - \mu \rvert}{\gamma}}$$


Laplace Distribution

The curve of probability density function of exponential distribution  has a sharp bend at x=0.

Probability density function of exponential distribution is given by

$$p(x;\lambda)=\lambda \mathbf{1}_{x\geq0} e^{-\lambda x}$$

For all negative values of x, $\mathbf{1}_{x\geq0}$ an indicator function assigns zero probability.

Exponential Distribution

Inorder to showcase a probability distribution where the mass can be concentrated at a single point,the probability density function is defined by dirac delta function $\delta(x)$.

The dirac delta function is zero everywhere except at x=0 and it has an area of one.

p{x) = $\delta(x-\mu)$,this equation explains that the function p(x) is having infinitemosuly large value at $x=\mu$ with unit area


Dirac Distribution

Empirical Distribution

Mixture distribution is a combination of several other probability distributions.

Mixture Distribution


There are some functions used while describing probability distribution in deep learning models they are
- Logistic Sigmoid
- Softplus Function


Properties of Common Functions

A prior probability distribution is a probability distribution over the parameters of a model that encodes our beliefs about what models are reasonable, before we have seen any data.

prior probability distribution

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.

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