# Lecture Notes
📗 Pseudorandom Numbers
➩ A random sequence can be generated by a recurrence relation, for example, \(x_{t+1} = \left(a x_{t} + c\right) \mod m\).
➩ \(x_{0}\) is called the seed, and if \(a, c, m\) are large and unknown, then the sequence looks random.
➩
numpy.random uses a more complicated PCG generator, but with a similar deterministic sequence that looks random:
Link 📗 Discrete Distributions
➩ A discrete distribution takes on finite or countably infinite number of values (for example, 0, 1, 2, ...).
➩ The distribution can be summarized in a table containing the probability that it takes on each value, for example \(\mathbb{P}\left\{X = a\right\}, a = 0, 1, 2, ...\).
➩ The probabilities should be non-negative and sum up to one.
📗 Continuous Distributions
➩ A continuous distribution takes on uncountably infinite number of values (for example, real numbers between 0 and 1).
➩ The distribution can be summarized as a probability density function \(f\left(x\right)\) with the property that \(\mathbb{P}\left\{a \leq X \leq b\right\} = \displaystyle\int_{a}^{b} f\left(x\right) dx\).
➩ The density function should be non-negative and integrates to 1.
➩ For a continuous \(X\), \(\mathbb{P}\left\{X = x\right\} = 0\) always has 0 probability of taking on any specific value.
Random Variable Examples
➩ Code for discrete distribution examples:
Notebook.
➩ Code for continuous distribution examples:
Notebook.
TopHat Discussion
➩ How to generate uniform random vectors within a circle or in a unit simplex?
➩ Code for uniform distribution in a circle or simplex:
Notebook.
Additional Example
➩ What is the correlation between \(\left(X, Y\right) \sim \text{\;MultivariateNormal\;} \left(\begin{bmatrix} 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \right)\)?
➩ The variance of \(X\) is \(\sigma_{X}^{2} = 1\), so the standard deviation is \(\sigma_{X} = 1\).
➩ The variance of \(Y\) is \(\sigma_{Y}^{2} = 4\), so the standard deivation is \(\sigma_{Y} = 2\).
➩ The covariance between \(X\) and \(Y\) is \(c_{X Y} = 2\), so the correlation is \(\rho_{X Y} = \dfrac{c_{X Y}}{\sigma_{X} \sigma_{Y}} = \dfrac{2}{1 \cdot 2} = 1\).
➩ Therefore, \(X\) and \(Y\) are perfectly positively correlated, i.e. all random pairs \(\left(X, Y\right)\) will appear on the same line.
Notes and code adapted from the course taught by Yiyin Shen
Link and Tyler Caraza-Harter
Link