Types of Continuous Probability Distributions

Here are some common types used in Machine learning,

Normal Distribution (Bell Curve) or Gaussian Distribution:

The Normal Distribution, sometimes referred to as the Gaussian Distribution, is a bell-shaped, symmetrical basic continuous probability distribution. Two factors define it: the standard deviation (σ), which indicates the distribution’s spread or dispersion, and the mean (μ), which establishes the distribution’s

• Parameters: Mean (μ) and standard deviation (σ)

For a random variable x, it is expressed as,

[Tex]f(x) =\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( -\frac{1}{2} \left( \frac{x – \mu}{\sigma} \right)^2 \right)[/Tex]

Note: The shape of the Normal Distribution is such that about 68% of the values fall within one standard deviation of the mean (μ ± σ), about 95% fall within two standard deviations (μ ± 2σ), and about 99.7% fall within three standard deviations (μ ± 3σ).

Uniform Distribution:

The Uniform Distribution is a continuous probability distribution where all values within a specified range are equally likely to occur.

• Parameters: Lower bound (a) and upper bound (b).
• The mean of a uniform distribution is [Tex]\mu = \frac{a + b}{2} [/Tex]and the variance is [Tex]\sigma^2 = \frac{(b – a)^2}{12} [/Tex]

It is expressed as:

[Tex]f(x) = \frac{1}{b – a} \quad \text{for } a \leq x \leq b [/Tex]

Exponential Distribution:

The exponential distribution is a continuous probability distribution that represents the duration between occurrences in a Poisson process, which occurs continuously and independently at a constant average rate.

• Parameter: Rate parameter (λ).
• The mean of the exponential distribution is 1/λ, and the variance is [Tex]1/λ^2.[/Tex]

For a random variable x, it is expressed as

[Tex]f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0 [/Tex]

Chi-Squared Distribution:

The Chi-Squared Distribution is a continuous probability distribution that arises in statistics, particularly in hypothesis testing and confidence interval estimation.

• It is characterized by a single parameter, often denoted as k ,which represents the degrees of freedom.
• The mean of the Chi-Squared Distribution is k and the variance is 2k.

For a random variable x, it is expressed as

[Tex]f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} \left( \frac{x}{2} \right)^{k/2 – 1} e^{-x/2} [/Tex]

Machine learning relies heavily on probability distributions because they offer a framework for comprehending the uncertainty and variability present in data. Specifically, for a given dataset, continuous probability distributions express the chance of witnessing continuous outcomes, like real numbers.