try to solve some of the problems given in the documentation

• What is normal distribution
• What is univariate, bivariate, and multivariate analysis
• what is bivariate normal distribution
• Marginal PDF

• Python
• JAX library
• matplotlib

Univariate: Using only one class of feature for generating the output

y = theta1*x + theta2

Where thetas are weight and bias

Bivariate: Using two different class of features for generating the output

z = theta1*x + theta2*y + theta3

This multivariate analysis help us to find the correlation between the features

Also known as gaussian distribution Continuous probability distribution for real-valued random variable Probability destity function

• Height of people
• Prices of shares in stock market
• Income distribution in ecomnomy
• Student average marks

jax.random.multivariate_normal(key, mean, cov, shape=None, dtype=<class 'numpy.float64'>, method='cholesky')   -> array

It is a matrix decomposistion method to factorize a matrix into a product of metrices It is a decomposition of a Hermitian (positive-definite) matrix into the product of a lower triangular matrix and its conjugate transpose. A = [L][L]T

[[4, 12, -16] 
 [12, 37, -43]
 [-16, -43, 98]
]
-----------------------
[2 0 0]            [2 6 -8]
[6 1 0]     *      [0 1 5]
[-8 5 3]           [0 0 3]

$$ \begin{aligned} & \text { }\\ &\begin{array}{cc} \hline \hline \text { Variance } & \text { Covariance } \\ \hline T(x{_i}-\mu)^2 & T((x{_1}{_i}-\mu{_1})(x{_2}{_i}-\mu{_2}))\\ % 2 & 47 & 877 & 230 \\ % 3 & 31 & 25 & 415 \\ % 4 & 35 & 144 & 23656 \\ % 5 & 45 & 300 & 556 \\ \hline \end{array} \end{aligned} $$

$$ \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix} , \begin{pmatrix} \sigma^2 {_1} & \sigma_1\sigma_2\\ \sigma_2\sigma_1 & \sigma^2 {_2} \end{pmatrix} \right) $$

• Variance refers to the spread of a data set around its mean value.
• covariance refers to the measure of the directional relationship between two random variables.

$$f\left(x\right)=\frac{\exp \left(\frac{-1}{2}(X-\mu)^{T} \Sigma^{-1}(X-\mu)\right)}{2 \pi|\Sigma|^{\frac{1}{2}}}$$