Author: marmara

Multivariate Distributions Random Vectors Definition: A vector $X$ is called a random vector if it has an associated pdf $f(.)$ \begin{eqnarray} X =\left[ \begin{array}{c} x_1\\…

Beta and Dirichlet Distributions Beta Distribution Beta distribution is frequently used as a prior. Dirichlet Distribution and Dirichlet Process Dirichlet distribution is a multinomial generalization…

Sample mean, sample variance and Bessel’s correction Sample mean is defined as \begin{eqnarray} \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i \end{eqnarray} and sample variance is defined as \begin{eqnarray}…

Snedecor’s F-distribution These two distributions are extremely important in statistics, as they underly the t-test and F-test. Here we simply define these distributions without touching…

Interarrival times for Poisson distribution: Exponential distribution \begin{eqnarray} f(x) = \lambda e^{-\lambda x} \end{eqnarray} Its moment is \begin{eqnarray} M(s) = \frac{1}{1-\frac{s}{\lambda}} \end{eqnarray} Higher order Interarrival…

Exponential Family of distributions The probability distributions we have discussed in the previous chapter (and many other frequently used ones) are members of the exponential…

Bernoulli Trials A Bernoulli trial is basically a single coin toss or a dice roll. It generates an experimental value for a discrete random variable…

wikipedia example If we have a pdf $f_X(x)$, its moment generating function is defined as \begin{eqnarray} M_X(t)=E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f_X(x) dx \end{eqnarray} On the…

Let $X$ be a random variable with pdf $f_X(x)$, and let $X$ be defined in the domain $x_0 \leq X \leq x_1$. Let $Y=g(X)$. Naturally,…