Some common random variables
- Discrete random variables
- Bernoulli(p)
- Binomial(n, p)
- Geometric(p)
- Poisson(\(\lambda\)): \(p(x) = e^{-\lambda}\frac{\lambda^x}{x!}\), non-negative integers
- Continuous random variables
- Uniform(a,b)
- Exponential(\(\lambda\)): \(f(x) = \lambda e^{-\lambda x}, x\geq 0\); \(F(x) = e^{-\lambda x}, x\geq 0\)
- Normal(\(\mu\), \(\sigma^2\))
Comment: some simulation methods 1. Inverse CDF technique : \(X = F^{-1}(U), U\sim unif(0,1)\) 2. Box Muller method for generating Gaussian
Two random variables
Expectation & Covariance
\[Cov[x,y] = E[(x-E[x])(y-E[y])] = E[XY] - E[X]E[Y]\]
Properties: - Linearity of expectation - \(Var[X+Y] = Var[X] + Var[Y] + 2Cov[X,Y]\) - If independent, \(Cov[X,Y]=0\) - If Independet, \(E[XY] = E[X]E[Y]\)
Multiple random variables
Random vectors
Vectorized denotation: \(X\)
Covariance matrix: PSD and symmetric \[\Sigma = E[XX^T] - E[X]E[X]^T = E[(X-E[X])(X-E[X])^T]\]
Multivariate Gaussian distribution
\[X\in \mathbb{R}^n \sim N(\mu, \Sigma)\]
\[f_X(X;\mu;\Sigma) = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}}\exp{\bigg(-\frac12(X-\mu)^T\Sigma^{-1}(X-\mu)\bigg)} \]